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DEPARTMENT  OF  (COMMERCE 

U.  S.  COAST  AND  GEODETIC  SURVEY 


O.    1  I.    •rri 

MUPKUIXTKSDKNT 


ASTRONOMY 


DETERMINATION  OF  TIME,  LONGITUDE 
LATITUDE,  AND  AZIMUTH 


FIFTH  EDITION 


BY 


BOWIE 

Inspector  of  G-eodetio  WorU  and  Cliief  of  tlie  Computing  Division 
TJ.  S.  Coast  and  Geodetic  Sui^vey 


SPECIAL   PUBLICATION    No.  14 


WASHINGTON 

GOVERNMENT  PRINTING  OFFICB 
1917 


DEPARTMENT   OF   COMMERCE 

U.  S.  COAST  AND  GEODETIC  SURVEY 


O.    H.   TI 

SUPERINTENDENT 


ASTRONOMY 


DETERMINATION  OF  TIME,  LONGITUDE 
LATITUDE,  AND  AZIMUTH 


FIFTH  EDITION 


BY 


WILLIAM    BCTWIK 

Inspector  of  Geodetic  "Work  and  Chief  of  the  Computing  Division. 
TJ.  S.  Coast  and  G-eodetic  Survey 


SPECIAL  PUBLICATION    No.   14 


PRICE,  65  CENTS 

Sold  only  by  the  Superintendent  of  Documents,  Government  Printing  Office,  Washington,  t>.  C. 

WASHINGTON 

GOVERNMENT  PRINTING  OFFICE 
1917 


CONTENTS. 


Page. 

Introduction 5 

PART  I.— DETERMINATION    OF    TIME. 

General  remarks 7 

Transit  instrument 7 

Transit  micrometer '. 8 

Chronograph 11 

Theory  of  the  transit  instrument 13 

Adjustments  of  the  transit  instrument 14 

Transit  observations 17 

Computation  of  transit  observations: 

Usual  method  of  computing  time  set 20 

Second  method  of  computing  time  set 28 

Least  square  method  of  computing  time  set  when  azimuth  stars  are  observed 39 

Complete  least  square  method  of  computing  time  set 41 

Determination  of  instrumental  constants 43 

Discussion  of  errors 48 

Other  methods  of  determining  time 51 

The  vertical  circle 52 

Star  factors 60 

PART  II.— THE  DETERMINATION  OF  THE  DIFFERENCE  OF  LONGITUDE  OF  TWO  STATIONS. 

Introductory 78 

Program  and  apparatus  of  the  telegraphic  method 79 

Computation  of  difference  of  longitude  when  transit  micrometer  is  used 84 

Discussion  of  errors,  transit  micrometer  method 85 

Program  where  no  transit  micrometer  is  used 87 

Computation  of  difference  of  longitude  when  no  transit  micrometer  is  used 87 

Personal  equation 90 

Discussion  of  errors,  key  method 93 

Statement  of  costs 94 

Longitude  by  the  chronometric  method 95 

Computation  of  longitude,  chronometric  method 97 

Discussion  of  errors,  chronometric  method 100 

PART  III.— THE  DETERMINATION  OF  LATITUDE  BY  MEANS  OF  THE  ZENITH  TELESCOPE. 

Introductory 103 

Instructions  for  latitude  work 103 

Instruments 104 

Adjustment  of  instruments 106 

Latitude  observations 107 

Computation  of  latitude Ill 

Apparent  places 116 

Corrections 117 

Combination  of  results 119 

Instrumental  constants •.-. 124 

Computation  of  micrometer  value 126 

Reductions  for  elevation  and  pole  variation 130 

Discussion  of  errors 132 

Economics  of  latitude  observations 135 

PART  IV.— THE  DETERMINATION  OF  THE  ASTRONOMIC  AZIMUTH  OF  A  DIRECTION. 

General  remarks 138 

Primary  azimuth 138 

Instruments 139 

General  considerations 142 

General  formula 143 

3 


4  CONTENTS. 

Page. 
PART  IV.— THE  DETERMINATION  OF  THE  ASTRONOMIC  AZIMUTH  OF  A  DIRECTION— Contd. 

Direction  method 145 

Method  of  repetitions 153 

Micrometric  method 155 

Discussion  of  errors 158 

Statement  of  costs 160 

Azimuth  from  time  observations 160 

Correction  for  elevation  of  mark  and  variation  of  the  pole 164 

Table  of  log  --L_ 165 

1  — a 

Index 175 

TABLES. 

Diurnal  aberration  («) 24 

For  use  in  computation  of  incomplete  transits 32 

Intervals  of  lines  of  transit  No.  18  from  mean  line 33 

Weights  for  incomplete  transits,  eye  and  ear  observations 36 

Weights  for  incomplete  transits,  chronographic  observations 38 

Relative  weights  to  transits  depending  on  the  star's  declination 39 

Refraction 58 

Sun's  parallax 60 

Star  factors 62 

Relative  personal  equation 92 

Correction  to  latitude  for  differential  refraction 118 

Correction  to  latitude  for  reduction  to  meridian 119 

Correction  for  curvature  of  apparent  path  of  star  in  computation  of  micrometer  value 127 

Reduction  of  latitude  to  sea  level 131 

Curvature  correction 150 

2  ^  *  T.  .  151 
sin  \" 

Logj-L..  165 

ILLUSTRATIONS. 

1.  Large  portable  transit  (equipped  with  transit  micrometer) 8 

2.  Broken  telescope  transit 8 

3.  Meridian  telescope 8 

4.  Transit  micrometer 10 

5.  Transit  micrometer 11 

6.  Chronograph 12 

7.  Portion  of  chronograph  record 13 

8.  Vertical  circle - 52 

9.  Nomogram  for  obtaining  star  factors 60 

10.  Arrangement  of  electrical  connections,  telegraphic  longitude — transit-micrometer  method 80 

11.  Arrangement  of  electrical  connections,  telegraphic  longitude — key  method 81 

12.  Switchboard — telegraphic  longitude 82 

13.  Zenith  telescope 104 

14.  Observatory 106 

15.  Observatory 107 

16.  Observiag  tent 108 

17.  Observiag  tent 108 

18.  Twelve-inch  direction  theodolite 138 

19.  Seven-inch  repeating  theodolite 138 

20.  Four-inch  theodolite 138 

21.  Small  acetylene  signal  lamp 140 

22.  Large  acetylene  signal  lamp 141 

23.  Eighty-foot  signal 142 

24.  Wooden  pier  used  for  theodolite  and  zenith  telescope 142 

25.  Structure  for  elevating  signal  lamp  over  triangulation  station  used  as  mark 144 

26.  Structure  for  elevating  signal  lamp  over  triangulation  station  used  as  mark 144 

27.  Azimuth  mark 145 

28 .  Circum polar  stars 146 

29.  Diagram  showing  directions  to  triangulation  stations  and  Polaris 147 


DETERMINATION  OF  TIME,  LONGITUDE,  LATITUDE,  AND  AZIMUTH. 


By  WILLIAM  BOWIE, 

Inspector  of  Geodetic  Work  and  Chief  of  the  Computing  Division,  U.  S.  Coast  and  Geodetic  Survey. 


INTRODUCTION. 

From  time  to  tune  during  many  years  publications  have  been  issued  describing  the 
instruments  and  methods  used  by  the  Coast  and  Geodetic  Survey  in  the  determination  of  time, 
longitude,  latitude,  and  azimuth.  The  general  aim  has  been  to  provide  a  working  manual 
which  would  serve  as  a  guide  to  the  observer  in  the  field  and  the  computer  in  the  office  in  carrying 
on  the  astronomic  work  of  the  Survey  in  a  systematic  manner.  The  exhaustion  of  previous 
editions  and  the  introduction  of  new  instruments  and  methods  have  made  necessary  the  suc- 
cessive editions,  in  each  of  which  much  has  been  repeated  from  the  preceding  one. 

The  edition  of  the  last  publication  is  now  exhausted,  which  gave  in  one  volume  descriptions 
of  the  instruments  and  methods,  and  was  entitled  "Determination  of  Time,  Longitude,  Latitude, 
and  Azimuth."  It  was  published  as  Appendix  No.  7,  Report  for  1898.  The  needs  of  the 
members  of  this  Survey  for  a  similar  manual,  and  requests  for  it  by  others,  make  it  desirable 
to  issue  the  present  and  fifth  edition. 

The  subject  matter  includes  most  of  that  in  the  fourth  edition,  with  a  number  of  changes, 
however.  Some  of  the  most  important  additions  to  the  previous  edition  arc :  The  determination 
of  time  and  longitude,  using  the  transit  micrometer;  the  description  of  the  transit  micrometer; 
determination  of  time  with  the  vertical  circle  for  use  in  connection  with  azimuth  observations; 
a  description  of  the  method  of  observing  azimuth  coincidently  with  horizontal  directions  in 
primary  triangulation ;  an  example  of  the  determination  of  an  azimuth  in  Alaska  with  a  transit 
equipped  with  a  transit  micrometer;  examples  of  the  records  and  computations  in  the  different 
classes  of  work,  as  actually  made  at  present  by  the  Survey;  and  statements  of  the  field  cost 
of  the  different  classes  of  work.  A  number  of  new  illustrations  have  been  added. 

The  writer  takes  pleasure  in  acknowledging  here  his  indebtedness  to  Mr.  H.  C.  Mitchell, 
Mr.  C.  R.  Duvall,  and  several  other  members  of  the  Computing  Division  who  assisted  in  preparing 
this  edition.  The  material  is  principally  the  work  of  former  Assistant  C.  A.  Schott,  who 
prepared  the  first  three  editions,  and  of  former  Assistant  John  F.  Hayford,  who  prepared  the 
fourth  edition. 

It  has  not  been  deemed  necessary  to  insert  the  derivation  of  formulae,  except  in  the  few 
rare  cases  in  which  such  derivation  can  not  be  found  readily  in  textbooks  on  astronomy.  For 
general  developments  the  reader  is  therefore  referred  to  Chauvenet's  Astronomy,  to  Doolittle's 
Practical  Astronomy,  and  to  Hayford's  Geodetic  Astronomy.  The  last-mentioned  book  and 
the  fourth  edition  of  this  publication  appeared  about  the  same  time,  and  as  they  were  by  the 
same  author  it  is  natural  that  some  of  the  text  is  identical  in  the  two.  Much  of  this  publication 
was  copied  from  the  fourth  edition  without  change,  and  some  portions  are  necessarily  identical 
with  the  corresponding  parts  of  Prof.  Hayford's  textbook. 

In  addition  to  this  manual  on  geodetic  astronomy,  the  American  Ephemeras  and  Nautical 
Almanac  for  the  year  of  observation  will  be  required  in  time  and  azimuth  work,  and  the  Boss 
Preliminary  General  Catalogue  of  6188  stars,  together  with  the  Cape  Tables,  by  Finlay,  in  latitude 
determinations. 

WILLIAM  BOWIE, 

Inspector  of  Geodetic  Work,  Chit f  of  the  Computing  Division. 

5 


PART    I. 

DETERMINATION  OF  TIME. 


GENERAL  REMARKS. 

This  part  deals  almost  exclusively  with  the  portable  transit  instrument  in  its  several  forms 
as  used  in  the  Coast  and  Geodetic  Survey,  and  when  mounted  in  the  plane  of  the  meridian  for 
the  purpose  of  determining  local  sidereal  time  from  observations  of  transits  of  stars,  in  connection 
with  an  astronomic  clock  or  chronometer  regulated  to  sidereal  time.  The  use  of  this  instrument 
when  mounted  in  the  vertical  plane  of  a  close  circumpolar  star  out  of  the  meridian  is  not  recom- 
mended on  account  of  the  greater  complexity  both  in  field  and  office  work,  as  compared  with  the 
usual  method  herein  discussed,  especially  when  one  considers  the  ease  with  which  a  transit  may 
be  placed  approximately  in  the  meridian.  (See  p.  16.)  The  observations  are  made  either  by  the 
method  of  "eye  and  ear,"  or  by  chronographic  registration.  The  latter  method  is  used  exclu- 
sively for  all  telegraphic  longitude  work  and  in  making  time  observations  for  determining  the 
periods  of  the  pendulums  in  gravity  determinations.  In  using  the  first  method  the  observer 
will,  of  course,  mark  his  own  time;  that  is,  he  will  pick  up  the  beats  of  the  chronometer  and 
carry  them  forward  mentally  up  to  the  time  of  transit  of  the  star,  which  he  will  estimate  to 
the  nearest  tenth  of  a  second.  In  using  the  second  method  the  chronograph  record  will  be 
produced  in  one  of  two  ways:  First,  when  the  observer  sees  the  star  bisected  by  a  line  of  the 
diaphragm  he  will  press  an  observing  key  (break-circuit)  held  in  his  hand  and  cause  a  record  of 
that  instant  to  appear  on  the  chronograph  sheet;  or,  second,  he  will  follow  the  star  across  the 
field  of  the  telescope  with  the  movable  wire  of  the  transit  micrometer,  the  star  being  continuously 
bisected  as  nearly  as  possible  by  the  wire,  and  the  record  on  the  chronograph  sheet  will  be  made 
automatically  by  the  make-circuit  device  of  the  micrometer. 

DESCRIPTION  OF  LARGE     PORTABLE     TRANSIT. 

Several  sizes  of  portable  transits  are  used  in  this  Survey.  The  largest  and  oldest  ones, 
made  by  Troughton  &  Simms,  of  London,  were  intended  for  use  exclusively  on  the  telegraphic 
determinations  of  longitude,  but  in  1888  a  slightly  smaller  t}rpe  of  transit  (described  below)  was 
made  at  the  Survey  office,  and  has  been  used  very  extensively  since  that  time  on  the  same  class 
of  work  as  the  largest  type.  The  smallest  type  of  transit,  known  as  the  meridian  telescope 
(described  on  p.  8),  is  used  in  the  determination  of  the  local  time  needed  while  observing 
astronomic  azimuths  and  latitudes,  and  for  other  purposes.  In  the  hands  of  skillful  observers 
the  instruments  used  for  longitude  determinations  give  results  which  compare  favorably  with 
the  results  obtained  with  the  much  larger  transits  usually  employed  at  astronomic  observatories, 
where  special  difficulties  are  encountered  in  consequence  of  strains  or  temporary  instability  of 
the  instrument  due  to  reversal  of  axis,  and  the  more  serious  effect  of  flexure.  In  case  of  necessity, 
and  when  an  approximate  degree  of  accuracy  suffices,  any  theodolite  or  altazimuth  instrument 
may  be  converted  temporarily  into  and  used  as  an  astronomic  transit. 

Illustration  No.  1  shows  Transit  No.  18,1  one  of  the  second-sized  portable  transits  made 
in  the  Survey  office  in  1888.  It  has  a  focal  length  of  94  cm.  and  a  clear  aperture  of  76  mm. 
The  magnifying  power  with  the  diagonal  eyepiece  ordinarly  used  is  104  diameters.  It  is  provided 
with  a  convenient  reversing  apparatus,  by  means  of  which  it  can  be  reversed  without  lifting  the 

1  For  a  full  description  of  this  instrument,  see  Appendix  9,  Report  for  1889,  by  Edwin  Smith,  Assistant. 


8  U.  S.  COAST  AND  GEODETIC   SURVEY   SPECIAL  PUBLICATION   NO.   14. 

telescope  by  hand.  The  value  of  one  division  (  =  2  mm.)  of  the  striding  level  is  1".35.  The 
setting  circles  are  4  inches  in  diameter,  are  graduated  to  20'  spaces,  and  arc  read  by  verniers  to 
single  minutes. 

Until  about  1905  this,  as  well  as  the  other  transits  of  the  Coast  and  Geodetic  Survey,  was 
supplied  with  a  glass  diaphragm,  but,  with  the  adoption  of  the  transit-micrometer,  the  glass 
diaphragms  were  discarded.  The  glass  diaphragm  carries  two  horizontal  lines  which  are  simply  to 
define  the  limits  within  which  all  observations  should  be  made,  and  13  vertical  lines,  11  of  which 
are  used  in  making  time  observations  with  the  chronograph  and  observing  key  and  5  of  which 
(longer  than  the  others)  are  used  in  making  eye  and  ear  observations.  The  shortest  time  interval 
between  lines  for  chronographic  observations  is  about  2£  seconds  and  for  eye  and  ear  observa- 
tions about  10  seconds.  The  transit  micrometer  and  its  use  are  described  below. 

Transit  No.  18  is  provided  with  a  sub-base  which  is  firmly  secured  to  the  supporting  pier. 
The  transit  proper  is  supported  on  this  sub-base  by  three  foot  screws.  At  the  left  of  the  base 
in  the  illustration  is  shown  a  pair  of  opposing  screws  which  serve  to  adjust  the  instrument  in 
azimuth.  One  of  these  screws  carries  a  graduated  head  which  enables  one  to  set  the  instrument 
very  nearly  in  the  meridian  as  soon  as  the  azimuth  error  is  known. 

This  instrument  may  serve  as  a  typical  illustration  of  the  class  of  large  portable  transits. 

The  broken  telescope  transit,  like  that  shown  in  illustration  NQ.  2,  has  been  used  with 
marked  success  by  other  countries.  This  instrument  may  also  be  used  in  the  determination  of 
latitude  by  the  Talcott  method.  This  manual  can  be  used  with  either  type  of  instrument  (broken 
or  straight  telescope) . 

DESCRIPTION   OF  MERIDIAN   TELESCOPE. 

Certain  instruments  are  known  in  this  Survey  as  meridian  telescopes.1  They  are  fitted 
both  for  time  observations  and  for  latitude  observations  by  the  Horrebow-Talcott  method 
(see  p.  103)  and  are  provided  with  a  frame  which  may  be  folded  up  for  convenience  in  transpor- 
tation. Illustration  No.  3  shows  Meridian  Telescope  No.  13,  which  may  serve  as  an  illustration 
of  the  type  of  smaller  instruments  used  for  time  observations  in  this  Survev. 

This  telescope  has  a  focal  length  of  66  cm.,  a  clear  aperture  of  5  cm.,  and  a  magnifying 
power  of  72  diameters.  The  value  of  one  division  (  =  2  mm.)  of  the  striding  level  is  about  2J". 
During  time  observations  the  telescope  is  reversed  by  hand;  during  latitude  observations  it  may 
be  reversed  by  turning  the  upper  half  of  the  double  base  on  the  lower  half.  One  of  the  two  setting 
circles  carries  a  delicate  level  for  use  in  making  latitude  observations,  and  the  eyepiece  is  fitted 
with  a  micrometer  for  measuring  differences  of  zenith  distance,  in  addition  to  the  diaphragm 
carrying  fixed  vertical  lines  for  use  in  making  time  observations.  On  one  side  of  the  base 
(the  left-hand  side  in  the  illustration)  is  a  slow-motion  screw  for  accurate  adjustment  in  azimuth. 

THE  TRANSIT  MICROMETER. 

The  transit  micrometer  is  a  form  of  registering  micrometer  placed  with  its  movable  wire  in 
the  focal  plane  of  an  astronomic  transit  and  at  right  angles  to  the  direction  of  motion  of  the 
image  of  the  star  which  is  being  observed  at  and  near  meridian  transit.  Certain  contact  points 
on  the  micrometer  head  serve  to  make  an  electric  circuit  as  they  pass  a  fixed  contact  spring,  thus 
causing  to  be  recorded  upon  the  chronograph  sheet  each  separate  instant  at  which  the  microm- 
eter wire  reaches  a  position  corresponding  to  a  contact. 

The  transit  micrometer  in  use  on  the  transits  of  this  Survey  is  hand  driven  and  was  designed 
by  Mr.  E.  G.  Fischer,  Chief  of  the  Instrument  Division  of  the  Survey,  and  made  in  that 
division.  Much  of  the  following  description  is  copied  from  pages  458-460  of  Appendix  No.  8, 
Report  for  1904,  entitled  "A  test  of  the  transit  micrometer."  The  pages  referred  to  were  written 
by  Mr.  Fischer. 

1  See  Appendix  No.  7,  Report  for  1879,  for  a  "  Description  of  the  Davidson  Meridian  Instrument. " 


No.  1. 


LARGE  PORTABLE  TRANSIT  (EQUIPPED  WITH  TRANSIT  MICROMETER). 


No.  2. 


BROKEN   TELESCOPE  TRANSIT. 


No.  3. 


-»#-* 


MERIDIAN  TELESCOPE. 


DETERMINATION    OF   TIME.  9 

DESCRIPTION  OF  THE  HAND-DRIVEN  TRANSIT  MICROMETER,   MADE  FOR  COAST  AND 

GEODETIC   SURVEY  TRANSIT  NO.   2. 

Before  considering  the  details  of  this  micrometer,  three  points  were  determined  upon 
as  being  essential  to  insure  accurate  and  decisive  action,  durability,  and  convenience  in  reading 
the  chronograph  record  made  by  it. 

First,  it  was  decided  that  the  mechanism  of  the  slide  carrying  the  wire  should  be  of  the 
form  in  which  the  screw  is  mounted  in  bearings  at  the  extreme  ends  of  the  box  or  case  holding 
the  slide,  the  micrometer  head  being  fast  upon  the  end  of  the  screw  projecting  from  the  box, 
because  this  insures  greater  stability  under  the  side  stress  of  the  gears  connecting  the  screw 
with  the  handwheel  shaft  than  the  form  usually  employed  in  theodolite  and  ocular  micrometers, 
in  which  the  screw  is  fastened  to  the  slide  and  therefore  takes  part  of  whatever  play  there  may 
be  in  the  latter. 

Second,  it  was  decided  that  the  electric  recording  device  of  the  micrometer  should  be  of 
the  make-circuit  form,  transmitting  its  records  to  the  chronograph,  which  is  in  the  break-circuit 
of  the  chronometer,  through  a  relay.  This  permits  the  use  of  a  strong  current  through  the 
contact  points  of  the  micrometer  head,  and  therefore  a  minimum  of  pressure  upon  the  latter  by 
the  contact  spring. 

Third,  in  order  that  the  micrometer  transmit  no  records  except  those  made  within  an 
accepted  space  on  either  side  of  the  line  of  collimation  and  forming  the  observations  of  the  star 
transits  proper,  an  automatic  cut-out  must  be  provided. 

Illustrations  4  and  5  show  the  micrometer  with  draw  tube  and  eye  end  of  the  telescope.  The 
telescope  has  a  focal  length  of  115  cm.  and  an  aperture  of  77  mm.  It  is  of  the  straight  type  of 
the  same  general  form  as  that  shown  in  illustration  No.  1  of  Appendix  7  of  the  Report  for  1898. 
(Illustration  No.  1  of  this  publication.) 

The  micrometer  box  or  case  is  46  mm.  in  length  and  31  mm.  wide.  Within  it  and  near  to 
one  side  is  mounted  the  micrometer  screw.  Upon  the  latter  fits,  by  a  thread  and  cylindrical 
bearing,  a  rectangular  frame  forming  the  slide,  which  is  31  mm.  long  and  23  mm.  wide.  All 
play  or  lost  motion,  both  of  the  slide  upon  the  screw  and  the  screw  in  its  bearings,  is  taken 
up  by  means  of  a  helical  spring  within  the  box,  which,  pressing  from  the  inner  end  of  the  box 
against  the  slide  and  through  it  against  the  screw,  holds  the  latter  firmly  against  the  point  of  an 
adjustable  abutting  screw,  without  impeding  its  free  rotary  motion.  Upon  the  slide,  at  right 
angles  to  its  line  of  motion,  is  mounted  the  single  spider  thread,  which  is  used  for  bisecting  the 
star  during  its  passage  across  the  field.  Two  threads,  parallel  to  the  line  of  motion,  about  four 
time  seconds  apart,  and  mounted  against  the  inner  surface  of  the  box,  define  the  space  within 
which  the  observations  should  be  made.  A  short  comb  of  five  teeth,  with  distances  equal  to  one 
turn  of  the  screw  between  them,  is  also  provided  and  indicates  the  four  whole  turns  of  the  screw 
within  which  the  observations  are  to  be  made.  The  diameter  of  the  field  of  view  through  the 
Airy  diagonal  eyepiece,  which  has  an  equivalent  focal  length  of  12  mm.,  is  something  over 
24  turns  of  the  screw,  thus  giving  a  space  of  fully  10  turns  of  the  screw  on  each  side  of  the  4 
turns  in  the  center  of  the  field. 

That  portion  of  the  micrometer  screw  which  projects  through  the  box  has  the  micrometer 
head  fitted  upon  it  and  secured  in  position  by  a  clamp  nut.  The  cylindrical  surface  of  this 
head,  graduated  at  the  edge  nearest  the  box  to  100  parts  (g,  illustration  No.  4),  also  carries 
near  its  opposite  edge  a  screw  thread,  t,  of  three  turns  with  a  pitch  of  1  mm.  and  a  diameter 
of  32  mm.  Sunk  into  the  outer  face  of  the  head  and  fitted  concentrically  with  it  is  a  thin 
metallic  shell,  which  has  fitted  upon  it  a  hollow  cylinder,  e,  made  of  ebonite,  6  mm.  long  and  26 
mm.  in  diameter.  Five  strips  of  platinum,  each  0.4  mm.  thick,  and  corresponding  to  the  12.5, 25.0, 
50.0,  75.0,  and  87.5  division  points  of  the  graduation,  g,  are  slotted  into  the  edge  of  the  ebonite 
cylinder  and  secured  in  such  manner  as  to  make  metallic  contact  with  the  micrometer  head 
proper,  and  through  it  with  the  screw,  micrometer  box,  telescope  and  telescope  pivots,  and  the 
iron  uprights  of  the  transit.  By  releasing  the  clamp  nut  within  the  ebonite  ring  the  graduated 


10  U.   S.   COAST  AND  GEODETIC   SURVEY  SPECIAL  PUBLICATION   NO.   14. 

head,  with  its  thread,  t,  can  be  adjusted,  in  a  rotary  sense,  in  relation  to  the  thread  of  the  screw, 
and  therefore  also  to  the  spider  thread  upon  the  slide.  At  the  same  time  the  position  of  the 
platinum  contact  strips  can  be  set  to  correspond  to  the  zero  of  the  graduation,  g,  which  latter 
is  read  by  the  index,  i,  illustration  No.  5. 

A  small  ebonite  plate,  p,  illustration  No.  4,  secured  to  the  micrometer  box,  carries  upon 
its  outer  end,  mounted  in  a  suitable  metal  block,  the  contact  spring,  s,  which  ends  in  a  piece 
of  platinum  turned  over  so  as  to  rest  radially  upon  the  ebonite  cylinder.  The  width  of  this 
piece  of  platinum  is  4  mm.,  and  its  thickness  that  of  the  contact  strips,  i.  e.,  0.4  mm.  A 
small  screw,  c,  illustration  No.  5,  serves  to  adjust  the  pressure  of  the  spring  upon  the  cylinder. 
Against  one  end  of  the  micrometer  box  is  fastened  a  small  bracket,  upon  which  is  centered  a 
small  worm  wheel,  w,  illustration  No.  4,  gearing  into  the  screw  thread,  t,  of  the  micrometer 
head.  It  has  40  teeth,  and  moves  1  tooth  for  each  turn  of  the  micrometer  head.  To  this  worm 
wheel  is  fastened  a  cup-shaped  cylinder,  r,  wliich  has  cut  into  its  rim  a  notch  or  depression 
with  sloping  ends  not  visible  in  the  illustrations.  A  small  steel  pin  in  the  end  of  the  lever,  I, 
rests  upon  the  edge  of  this  cup-shaped  cylinder.  The  other  end  of  the  lever,  I,  fitted  with  a 
small  ivory  tip,  presses  upon  the  end  of  the  contact  spring,  &,  which  is  mounted  upon  an  ebonite 
plate,  and  is  therefore  insulated  electrically  from  the  instrument.  When  the  small  steel  pin 
rests  upon  the  edge  of  the  cup-shaped  cylinder,  the  ivory  tip  presses  the  contact  spring  away 
from  the  platinum-tipped  screw,  a.  When,  however,  the  notch  or  depression  comes  below  the 
steel  pin,  the  contact  spring,  6,  is  free  to  press  against  the  platinum-tipped  screw,  thus  allowing 
the  flow  of  an  electric  current  through  the  coiled  wires,  m  and  n,  and  the  contact  spring,  s.  The 
length  of  the  notch  is  chosen  so  as  to  allow  the  circuit  to  be  closed  during  four  revolutions 
of  the  micrometer  head.  As  the  ends  of  the  notch  are  sloping,  it  will  be  seen  that  by  raising 
or  lowering  the  platinum-tipped  screw,  and  consequently  lowering  or  raising  respectively  the 
steel  pin  in  the  lever  I,  the  time  during  which  the  current  can  flow  can  be  made  to  correspond 
exactly  to  that  of  four  revolutions  of  the  micrometer  head.  But  it  is  also  important  that  the 
four  revolutions  during  which  the  current  can  flow  and  record  the  contacts  made  on  the  ebonite 
cylinder,  e,  are  those  disposed  symmetrically  about  the  zero  position  of  the  micrometer,  wliich 
indicates  the  meridian.  This  is  accomplished  for  adjustments  requiring  corrections  greater  than 
one  tooth  of  the  worm  wheel  w,  by  removing  the  latter  from  its  axis,  turning  and  replacing  it 
with  the  proper  tooth  engaging  the  screw  thread,  t.  The  adjustment  for  amounts  less  than 
that  of  one  tooth,  as  the  micrometer  is  now  arranged,  is  made  by  loosening  a  capstan-headed 
screw  (hidden  in  the  illustration  by  the  lever  1),  and  turning  to  right  or  left  the  two  screws  z,  thus 
moving  the  plate  carrying  the  lever  I,  until  the  small  steel  pin  at  the  end  of  lever  I  is  in  proper 
relation  to  the  notch  or  depression  in  the  cup-shaped  cylinder  r.  It  will  be  seen,  therefore, 
that  tlu's  arrangement  permits  of  the  motion  of  the  spider  thread  across  the  entire  field  without 
transnu'tting  records  to  the  chronograph,  except  during  the  four  revolutions  symmetrically 
disposed  about  the  line  of  collimation. 

Against  the  inner  face  of  the  micrometer  head  is  fastened  a  spur  wheel,  k,  illustration  No.  5, 
with  36  teeth  of  48  diametral  (inch)  pitch,  into  which  gears  the  wheel/,  with  72  teeth,  mounted 
on  the  handwheel  shaft,  d.  This  shaft  is  supported  by  arms  from  the  micrometer  box,  as  can 
readily  be  seen  from  illustration  No.  5.  The  handwheels  have  a  diameter  of  33  mm.,  are  1 16  mm. 
apart,  and  equidistant  from  the  middle  of  the  telescope,  allowing  ample  space  for  manipulating  in 
either  position  of  the  eyepiece. 

The  pitch  of  the  micrometer  screw  is  about  48.4  threads  per  centimeter,  or  123  per  inch. 
In  the  telescope  of  Transit  No.  2  the  angular  value  of  one  revolution  of  the  screw  is  2.5  equatorial 
time  seconds,  nearly.  As  the  gearing  of  the  handwheel  shaft  to  the  micrometer  screw  is  as  2 
to  1  it  follows  that  the  hands  must  produce  rotary  motion  of  one  revolution  in  about  5s  for  an 
equatorial  star. 

The  adjustment  for  collimation  is  made  by  means  of  two  nuts,  x,  illustration  No.  4,  upon 
a  small  screw  fastened  to  the  micrometer  box,  which  in  turn  is  mounted  by  dovetail  slides 
upon  a  short  flanged  cylinder,  y.  The  latter  is  fixed  in  position  by  the  screws,  h,  which,  when 
loosened,  also  permit  of  a  rotary  motion  for  adjusting  the  transit  wire  into  the  vertical.  Neither 


No  4. 


TRANSIT  MICROMETER. 


No.  5. 


TRANSIT  MICROMETER. 


DETERMINATION   OF   TIME.  11 

of  these  adjustments  will  disturb  the  rather  delicate  relations  between  the  zero  of  the  transit 
wire,  the  contact  breaks  upon  the  micrometer  head,  and  the  worm  wheel  with  its  electric  cut-out 
attachment. 

As  indicated  in  the  description  of  the  ebonite  head  with  its  five  platinum  contact  strips, 
the  instrument  itself  is  used  as  part  of  the  electric  conductor  forming  the  transit  circuit.  The 
relay  of  20  ohms  resistance  converts  the  makes  of  the  transit  circuit  into  breaks  in  the  chrono- 
graph circuit.  From  the  contact  spring,  6,  through  wire,  m,  connection  is  made  with  an  insu- 
lated binding  post  at  the  eye  end  of  the  telescope  tube,  from  which  a  wire  leads  along  the  tele- 
scope to  and  into  the  telescope  axis  and  within  the  latter  to  an  insulated  metal  cylinder  pro- 
jecting from  the  transit  pivot.  Each  of  the  wye  bearings  of  the  transit  has  fastened  to  it  an 
insulated  contact  spring,  which,  being  connected  with  an  insulated  binding  post  at  the  foot  of 
the  instrument,  establishes  the  circuit  whether  the  telescope  lies  in  either  an  east  or  west  posi- 
tion. Another  binding  post,  screwed  directly  into  the  iron  foot  of  the  transit,  affords  a  ready 
means  for  making  the  necessary  connection  to  begin  observations. 

It  is  necessary  to  use  both  hands  in  order  to  impart  to  the  wire  a  steady  motion.  As 
explained  above,  the  cut-out  device  allows  only  a  limited  portion  of  the  field  of  observation 
to  be  registered,  by  automatically  breaking  the  transit  circuit  while  the  wire  is  outside  the 
limits.  It  requires  four  complete  revolutions  of  the  micrometer  head  to  carry  the  wire  across  the 
field  of  record  and  as  there  are  five  contact  strips  on  the  micrometer  head,  the  complete  record 
of  the  observation  of  the  transit  of  a  given  star  consists  of  20  breaks  registered  on  the  chrono- 
graph sheet.  As  the  five  contact  strips  are  not  equally  spaced  around  the  head  of  the  microm- 
eter wheel,  it  follows  that  the  record  is  in  four  groups  of  five  observations  each.  This  facilitates 
the  reading  of  the  chronograph  sheet.  The  transit  of  an  equatorial  star  across  the  field  of 
record  occupies  only  about  10  seconds  of  time,  a  fact  which  makes  it  possible  to  observe  stars 
which  are  quite  close  together  in  right  ascension. 

Adjustments  of  the  transit  micrometer. — Before  using  the  transit  micrometer  it  should  be 
carefully  examined  to  see  that  there  is  no  loose  play  in  any  of  its  parts,  that  its  contact  strips 
and  contact  spring  are  clean  and  bright,  and  that  the  cut-out  attachment  permits  the  recording 
of  20  breaks  which  are  symmetrical  about  the  mean  position  of  the  micrometer  wire.  If  a 
symmetrical  record  is  not  obtained,  the  adjustment  must  be  made,  as  described  on  page  10. 

The  adjustment  of  the  micrometer  wire  for  collimation  and  verticality  are  described  on 
page  15,  under  the  heading  "Adjustment  of  the  transit  instrument." 

THE  CHRONOGRAPH. 

Illustration  No.  6  shows  the  form  of  chronograph  now  in  use  in  the  Survey.  The  train  of 
gears  seen  at  the  right  is  driven  by  a  falling  weight.  It  drives  the  speed  governor  (seen  above 
the  case  containing  the  gears),  the  cylinder  iipon  which  the  record  sheet  is  wound,  and  the 
screw  which  gives  the  pen  carriage  a  slow  motion  parallel  to  the  axis  of  the  record  cylinder. 
When  the  speed  governor  is  first  released,  the  speed  continually  increases  until  the  governor 
balls  have  moved  far  enough  away  from  the  axis  of  revolution  to  cause  a  small  projection  upon 
one  of  them  to  strike  a  small  hook.  This  impact  and  the  effect  of  the  friction  at  the  base  of 
the  weight  attached  to  the  hook  causes  the  speed  to  decrease  continually  until  the  hook  is  released. 
The  speed  then  increases  again  until  the  hook  is  engaged,  decreases  until  it  is  released,  and  so 
on.  The  total  range  of  variation  in  the  speed  is,  however,  surprisingly  small,  so  small  that 
in  interpreting  the  record  of  the  chronograph  the  speed  is  assumed. to  be  uniform  during  the 
intervals  between  chronometer  breaks.  The  speed  may  be  regulated  by  screwing  or  unscrewing 
the  movable  weights  which  are  above  the  governor  balls  and  attached  to  the  same  arm.  This 
moves  them  nearer  to  or  farther  from  the  axis,  and  thus  decreases  or  increases  the  critical  speed 
at  which  the  hook  is  engaged.  To  get  a  convenient  record  it  is  desirable  to  adjust  the  speed  so 
that  the  record  cylinder  makes  just  one  revolution  per  minute  with  the  ordinary  arrangement 
of  the  train  of  gears.  The  gears  may  also  be  changed  quickly  to  another  combination  which 
will  run  the  record  cylinder  at  double  speed.  This  will  require  additional  driving  weights. 


12  U.   S.   COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.    14. 

The  chronograph  circuit,  passing  through  the  coils  of  the  pen  magnet,  is  operated  by  a 
battery  of  two  dry  cells  in  series,  so  that  a  relatively  strong  spring  may  be  used  to  draw  the  pen 
armature  away  from  the  pen  magnet  when  the  circuit  is  broken.  This  insures  a  sharp  lateral 
movement  of  the  recording  pen,  which  is  attached  to  the  pen  armature,  on  the  breaking  of  the 
circuit,  and  a  correspondingly  sharp  offset  or  break  is  secured  in  the  helix  which  the  pen  traces  on 
the  drum. 

When  observations  are  made  on  the  lines  of  a  reticle,  an  observing  key  is  placed  in  the 
chronograph  circuit,  which  normally  keeps  the  circuit  closed,  and  breaks  it  only  when  the  key 
is  pressed  by  the  observer  as  the  star  is  bisected  by  each  of  the  lines  of  the  reticle. 

When  the  transit  micrometer  is  used,  the  transit  circuit,  passing  through  the  transit,  the 
micrometer  head  and  the  coils  of  the  transit  relay,  and  operated  by  two  dry  cells  in  series,  is 
connected  with  the  chronograph  circuit  through  the  points  of  the  transit  relay.  The  observing 
key  and  the  transit  circuit  with  its  relay  may  be  regarded  as  interchangeable,  as  either  one 
may  be  joined  into  the  chronograph  circuit  in  the  place  of  the  other. 

The  chronometer  circuit  is  operated  by  a  single  dry  cell,  and  passes  through  the  coils  of  a 
relay,  through  the  points  of  which  it  is  connected  with  the  chronograph  circuit.  Breaks  in  the 
chronometer  circuit  are  transmitted  into  breaks  in  the  chronograph  circuit  by  means  of  the 
chronometer  relay.  A  condenser  should  be  placed  in  the  circuit  across  the  terminals  of  the 
chronometer  to  prevent  sparking  and  consequent  injury  to  the  contact  points  of  the  break 
circuit  wheel  in  the  chronometer. 

The  strength  of  the  current,  the  tightness  of  the  spring  which  draws  back  the  pen  armature, 
the  distance  of  that  armature  from  the  magnet  core,  and  the  range  of  movement  of  the  armature 
must  all  be  adjusted  relatively  to  each  other  so  that  the  pen  will  furnish  a  neat  and  complete 
record  of  all  the  breaks  in  the  circuit.  The  driving  weight  must  be  heavy  enough  to  overcome 
all  friction  and  cause  the  governor  hook  to  be  engaged  frequently,  but  it  must  not  be  so  heavy 
as  to  cause  the  hook  to  be  carried  forward  continuously  after  it  is  once  engaged.  Where  a  transit 
micrometer  is  used  and  the  chronograph  circuit  is  broken  by  means  of  a  relay  placed  in  the 
transit  circuit,  this  relay  also  must  be  adjusted  to  produce  a  short  neat  break  of  the  chrono- 
graph circuit. 

In  operation  the  chronometer  breaks  the  circuit  automatically  every  second  (or  every  two 
seconds)  and  the  pen  records  the  breaks  upon  the  moving  record  sheet  at  equal  or  very  nearly 
equal  linear  intervals.  The  chronometer  is  usually  arranged  to  indicate  the  beginning  of  each 
minute  by  failing  to  make  a  break  for  the  fifty-ninth  second,  or  if  it  is  a  two-second  chronometer, 
by  making  a  break  for  the  fifty-ninth  second.  The  hours  and  minutes  may  be  identified  by 
writing  upon  some  point  of  the  record  sheet  the  corresponding  reading  of  the  face  of  the 
chronometer.  In  longitude  work  it  is  not  essential  to  have  the  hours  and  minutes  on  the 
chronograph  sheet  correspond  to  those  shown  on  the  face  of  the  chronometer.  It  is  customary 
to  mark  on  the  chronograph  sheet  such  hours  and  minutes  as  will  give  the  clock  a  correction 
of  less  than  one  minute,  which  is  equivalent  to  setting  the  chronometer  to  produce  that  reading. 

The  record  of  the  exact  time  of  the  transit  of  a  star  is  obtained  in  the  following  manner : 
Where  a  transit  micrometer  is  used  the  star  is  bisected  with  the  wire  of  the  micrometer  soon  after 
it  enters  the  field  of  view  of  the  telescope  (see  p.  18),  and  the  observer  endeavors  to  keep  the 
star  bisected  as  it  crosses  the  field.  As  the  wire  passes  the  various  positions  corresponding  to 
contacts  on  the  micrometer  head  the  transit  circuit  is  automatically  made,  and  through  the 
action  of  a  relay  it  automatically  breaks  the  chronograph  circuit  and  produces  a  record  on  the 
chronograph  sheet.  Where  an  observing  key  is  used  the  observer  breaks  the  chronograph 
circuit  directly  by  pressing  the  key  wliich  he  holds  in  his  hand ;  this  is  done  as  the  star  transits 
each  line  of  the  reticle.  In  each  case  the  position  of  the  additional  break  or  record  on  the  chro- 
nograph sheet,  with  reference  to  the  record  made  by  the  chronometer,  indicates  accurately  the 
chronometer  time  at  wliich  it  was  made,  the  chronograph  being  assumed  to  run  uniformly 
between  adjacent  chronometer  breaks.  (See  illustration  No.  7.)  To  read  the  fractions  of 
seconds  from  the  chronograph  sheet  one  may  use  either  a  glass  scale  on  wliich  converging  lines 
make  it  possible  to  divide  varying  lengths  of  seconds  into  10  equal  spaces,  or  a  small  linear 


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DETERMINATION   OF    TIME.  13 

rule,  so  divided  that  10  of  its  spaces  fit  closely  a  second's  interval  of  the  chronograph,  when 
the  chronograph  is  making  exactly  one  revolution  per  minute.  Some  of  the  chronographs  now 
in  use  in  the  Survey  are  so  constructed  that  when  in  perfect  adjustment  one  second  on  the 
record  will  be  exactly  1  cm.  in  length.  Such  a  record  may  be  easily  read  by  using  a  meter  scale. 
When  the  linear  scale  does  not  fit  the  chronograph  record  exactly  a  satisfactory  reading  is 
obtained  by  a  slight  shifting  of  the  scale  to  fit  the  adjacent  seconds  marks  as  the  transit  records 
are  successively  read.  This  linear  scale  is  much  preferred  to  the  glass  scale,  as  it  enables  one 
to  read  the  complete  record  for  a  star  with  one  setting  of  the  scale.  Also  by  placing  the  0 
mark  of  the  scale  on  an  even  10-second  mark  (0,  10,  20,  etc.)  immediately  preceding  the  stai's 
record,  not  only  the  fractional  part  of  the  second  may  be  read  at  once,  but  also  the  number 
of  the  second.  The  beginning  of  each  break  made  by  the  observer  and  by  the  chronometer  is 
the  exact  point  to  be  used  in  reading  the  chronograph  record,  the  break  of  the  circuit  being  sharp 
and  definite,  while  the  make  is  indefinite.  When  an  observing  key  is  used  and  11  breaks 
constitute  a  full  record  for  a  star,  the  star  transits  are  usually  read  from  the  record  sheet  to  the 
nearest  half-tenths  (0.05)  of  a  second;  when  a  transit  micrometer  is  used  and  20  obser- 
vations constitute  the  full  record  of  a  transit,  the  readings  are  made  to  the  nearest  tenth  (0.1) 
of  a  second  only.  In  longitude  work  it  is  customary  to  read  the  time  signals  to  the  nearest 
hundredth  (0.01)  of  a  second,  the  chronograph  then  being  run  at  double  speed.  There  will 
occasionally  be  a  slight  interference  between  the  chronometer  and  the  star  transit  record  caused 
by  overlapping,  but  the  time  of  the  observation  can  usually  be  identified  and  closely  estimated 
by  comparing  the  distances  between  the  successive  breaks. 

A  correction,  called  the  contact  correction,  is  sometimes  applied  to  the  chronograph  record 
of  transits  observed  with  a  micrometer  to  account  for  the  time  required  for  the  contact  spring  to 
cross  the  contact  strip  on  the  head  of  the  micrometer.  In  order  to  insure  a  satisfactory  record 
the  contact  strips  on  the  micrometer  are  given  material  width,  since  if  they  were  reduced  too 
much  there  would  be  an  occasional  skipping  of  a  record.  The  micrometer  wire  travels  from  a 
different  side  of  the  instrument  for  upper  and  lower  culminating  stars,  and  also  before  and 
after  reversal  of  the  telescope  in  its  wyes,  so  that  the  contact  spring  produces  a  record  sometimes 
from  one  edge  of  the  contact  strip  and  sometimes  from  the  other.  Theoretically,  the  proper 
reduction  would  be  to  correct  all  observations  for  one-half  the  movement  of  the  micrometer 
wire  from  the  beginning  of  the  contact  to  its  end.  This  may  be  measured  on  the  micrometer 
head.  The  micrometer  is  turned  very  slowly  until  the  armature  of  a  relay,  in  the  transit  circuit 
is  heard  to  make  the  circuit;  the  micrometer  head  is  then  read.  The  motion  is  continued 
until  the  armature  sounds  the  breaking  of  the  circuit,  and  the  micrometer  is  read  again.  The 
difference  between  the  two  readings  is  the  movement  of  the  wire  in  terms  of  divisions  on  the 
micrometer  head.  This  may  be  reduced  to  time  when  the  equatorial  value  of  the  micrometer 
division  is  known.  This  correction  is  always  plus,  since  the  middle  of  the  strip  must  always 
come  under  the  contact  spring  later  than  does  its  near  edge.  But  being  very  small  and  having 
nearly  the  same  effect  on  all  time  determinations  with  similar  instruments  it  is  without  appre- 
ciable effect  on  the  observed  differences  of  longitude.  Nor  is  this  correction  necessary  in  time 
determinations  for  gravity  observations  with  pendulums.  If  we  designate  the  contact  correction 
on  an  equatorial  star  for  any  transit  micrometer  as  n,  then  the  contact  correction  for  any  star 
is  n  sec  dorn  C,  where  C,  the  collimation  factor,  is  obtained  directly  from  the  table  on  pages  62-77, 
or  graphically  as  shown  in  illustration  No.  9.  The  equatorial  contact  correction  on  transit 
No.  18  is  0.008  second. 

THEORY  OF  THE   TRANSIT   INSTRUMENT. 

The  meaning  of  the  phrase  line  of  collimation  used  in  the  preceding  edition  of  this  publication 
vAppendix  No.  7,  of  1898)  is  adhered  to  in  the  present  publication.  The  line  of  collimation  may 
be  defined  as  the  line  through  the  optical  center  of  the  objective  and  the  middle  point  of  the 
mean  vertical  line  of  the  diaphragm  or  the  micrometer  wire  in  its  mean  position.  It  may  be 
considered  synonymous  with  the  pointing  line,  sight  line,  or  line  of  sight.  The  term  collimation 
axis  as  used  in  this  publication  may  be  defined  as  the  line  through  the  optical  center  of  the 


14  U.   S.   COAST  AND  GEODETIC   SUBVEY   SPECIAL  PUBLICATION   NO.   14. 

objective,  and  perpendicular  to  the  horizontal  axis  (axis  of  rotation)  of  the  telescope.  The 
line  of  collimation  and  collimation  axis  of  a  telescope  coincide  only  when  there  is  110  error  of 
collimation  hi  the  instrument. 

If  a  transit  instrument  were  in  perfect  adjustment  the  line  of  collimation  of  the  telescope 
would  be  at  right  angles  to  the  transverse  axis  upon  which  the  telescope  rotates,  and  that 
transverse  axis  would  be  horizontal  and  in  the  prime  vertical.  Under  these  circum- 
stances the  line  of  collimation  would  always  lie  in  the  meridian  plane,  and  local  sidereal  time 
at  the  instant  when  a  given  star  crossed  the  line  of  collimation  would  necessarily  be  the  same  as  the 
right  ascension  of  that  star.  The  difference  then  between  the  chronometer  time  of  transit  of 
a  given  star  across  the  line  of  collimation  and  the  right  ascension  of  that  star  would  be  the  error 
of  the  chronometer  on  local  sidereal  time.  Before  observing  meridian  transits  for  the  deter- 
mination of  time,  the  conditions  stated  in  the  first  sentence  of  this  paragraph  are  fulfilled  as 
nearly  as  possible  by  careful  adjustment  of  the  instrument.  The  time  observations  them- 
selves and  certain,  auxiliary  observations  are  then  made  in  such  a  manner  that  the  small  remain- 
ing errors  of  adjustment  may  be  determined,  and  the  observed  times  of  transit  are  corrected 
as  nearly  as  may  be  to  what  they  would  have  been  had  the  observations  been  made  with  a 
perfectly  adjusted  instrument.  The  observed  chronometer  time  of  transit  of  any  star  across 
the  line  of  collimation  as  thus  corrected  being  subtracted  from  the  right  ascension  of  that  star 
gives  the  correction  (on  local  sidereal  time)  of  the  chronometer  used  during  the  observations. 

ADJUSTMENTS  OF  THE  TRANSIT  INSTRUMENT. 

Let  it  be  supposed  that  observations  are  about  to  bo  commenced  at  a  new  station  at  which 
the  pier  and  shelter  for  the  transit  have  been  prepared.  (See  p.  105.)  By  daylight  make  the 
preparations  described  below  for  the  work'  of  the  night. 

By  whatever  .means  are  available  determine  the  approximate  direction  of  the  meridian 
and  mark  it  on  the  top  of  the  pier  or  by  an  outside  natural  or  artificial  signal.  Place  the 
sub-base  or  footplates  of  the  instrument  in  such  position  that  the  telescope  will  swing  closely  in 
the  meridian.  It  is  well  to  fix  the  sub-base  or  footplates  firmly  in  place  by  cementing  them 
to  the  pier  with  plaster  of  Paris  when  a  stone,  concrete,  or  brick  pier  is  used,  and  by  screws 
or  bolts  when  a  wooden  pier  is  used.  The  meridian  may  be  determined  with  sufficient  accuracy 
for  this  purpose  by  means  of  a  compass  needle,  the  magnetic  declination  being  known  and 
allowed  for.  A  known  direction  from  triangulation  or  from  previous  azimuth  observations 
may  be  utilized.  All  that  is  required  is  that  the  telescope  shall  be  so  nearly  in  the  meridian 
that  the  final  adjustment  will  come  within  the  scope  of  the  screws  provided  upon  the  instru- 
ment for  the  azimuth  adjustment. 

Set  up  the  instrument  and  inspect  it.  The  pivots  and  wyes  of  both  instrument  and  level 
should  be  cleaned  with  watch  oil,  which  must  be  wiped  off  to  prevent  its  accumulating  dust. 
They  should  be  carefully  inspected  to  insure  that  there  is  110  dirt  gummed  to  them.  The  lens 
should  be  examined  occasionally  to  see  that  it  is  tight  in  its  cell.  It  mav  be  dusted  off  witli  a 
camel's-hair  brush,  and  when  necessary  may  be  cleaned  by  rubbing  gently  with  soft,  clean 
tissue  paper,  first  moistening  the  glass  slightly  by  breathing  on  it. 

Focus  the  eyepiece  by  turning  the  telescope  up  to  the  sky  and  moving  the  eyepiece  in 
and  out  until  that  position  is  found  in  which  the  most  distinct  vision  is  obtained  of  the  micrometer 
wire.  If  any  external  objects  are  visible  through  the  eyepiece  in  addition  to  the  micrometer 
wire  seen  projected  against  a  uniform  background  (the  sky,  for  example)  the  eye  will  attempt, 
in  spite  of  its  owner,  to  focus  upon  those  objects  as  well  as  upon  the  micrometer  wire  and  the 
object  of  the  adjustment,  namely,  to  secure  a  focus  corresponding  to  a  minimum  strain  upon  the 
eye,  will  be  defeated  to  a  certain  extent. 

Focus  the  objective  by  directing  the  teloscope  to  some  well-defined  object,  not  less  than  a 
mile  away,  and  changing  the  distance  of  the  objective  from  the  plane  in  which  the  micrometer 
wire  moves  until  there  is  no  apparent  change  of  relative  position  (or  parallax)  of  the  micrometer 
wire  and  the  image  of  the  object  when  the  eye  is  shifted  about  the  front  of  the  eyepiece.  The 


DETERMINATION   OF   TIME.  15 

object  of  the  adjustment,  namely,  to  bring  the  image  formed  by  the  objective  into  coincidence 
with  the  micrometer  wire  is  then  accomplished.  If  the  eyepiece  has  been  properly  focused  this 
position  of  the  objective  will  also  be  ths  position  of  most  distinct  vision.  The  focus  of  the 
objective  will  need  to  be  inspected  at  night,  using  a  star  as  the  object,  and  corrected  if  necessary. 
Unless  the  focus  is  made  nearly  right  by  daylight  none  but  the  brightest  stars  will  be  seen  at  all 
at  night  and  the  observer  may  lose  time  trying  to  learn  the  cause  of  the  trouble.  If  the  objective 
is  focused  at  night  a  preliminary  adjustment  should  be  made  on  a  bright  star  and  the  final 
adjustment  on  a  faint  star,  as  it  is  almost  impossible  to  get  a  very  sharp  image  of  a  large  star. 
A  planet  or  the  moon  is  an  ideal  object  on  which  to  focus  the  objective.  A  scratch  upon  the  draw- 
tube  to  indicate  its  approximate  position  for  sidereal  focus  will  be  found  a  convenience.  After 
a  satisfactory  focus  has  been  found  the  drawtube  is  clamped  in  position  with  screws  provided 
for  that  purpose. 

Methods  exactly  similar  to  those  described  in  the  two  preceding  paragraplis  are  employed 
in  focusing  the  eyepiece  and  objective  when  a  diaphragm  is  used  instead  of  the  micrometer. 

If  unusual  difficulty  is  had  with  the  illumination  at  night,  it  is  advisable  to  remove  the 
eyepiece  and  look  directly  at  the  reflecting  mirror  in  the  telescope  tube.  The  whole  surface  of  the 
mirror  should  be  uniformly  illuminated.  If  tliis  is  not  the  case,  the  mirror  should  be  rotated 
until  a  satisfactory  illumination  is  obtained.  Occasionally  the  mirror  must  be  removed  from  the 
telescope  and  its  supporting  arm  bent  in  order  to  make  the  reflected  rays  of  light  approximately 
parallel  with  the  tube  of  the  telescope. 

Adjust  the  striding  level  in  the  ordinary  manner,  placing  it  on  the  pivots  direct  and  reversed. 
If  the  level  is  already  in  perfect  adjustment  the  difference  of  the  two  east  (or  west)  end  read- 
ings will  be  zero  for  a  level  numbered  in  both  directions  from  the  middle,  or  the  sum  of  the  two 
east  (or  west)  end  readings  will  be  double  the  reading  of  the  middle  of  the  tube  for  a  level  num- 
bered continuously  from  one  end  to  the  other.  The  level  must  also  be  adjusted  for  wind.  In 
other  words,  if  the  axis  of  the  level  tube  is  not  parallel  to  the  line  joining  the  wyes,  the  bubble 
will  move  longitudinally  when  the  level  is  rocked  back  and  forth  on  the  pivots.  The  adjustment 
for  wind  is  made  by  means  of  the  side  adjusting  screws  at  one  end  of  the  level.  To  adjust  for 
wind,  move  the  level  forward  and  then  back  and  note  the  total  movement  of  the  bubble.  The 
wind  will  be  eliminated  by  moving  the  bubble  back  one-half  of  the  total  displacement  by  means 
of  the  side  adjusting  screws.  Then  test  again  for  wind,  and  repeat  adjustment  if  necessary. 
In  placing  the  level  upon  the  pivots  it  should  always  be  rocked  slightly  to  insure  its  being  in  a 
central  position  and  in  good  contact. 

Level  the  horizontal  axis  of  the  telescope. — This  adjustment  may,  of  course,  be  combined  with 
that  of  the  striding  level. 

Test  the  verticality  of  the  micrometer  wire  (or  of  the  lines  of  the  diaphragm)  by  pointing 
on  some  well-defined  distant  object,  using  the  apparent  upper  part  of  the  wire  (or  of  the  middle 
line  of  the  diaphragm).  Rotate  the  telescope  slightly  about  its  horizontal  axis  until  the  object 
is  seen  upon  the  apparent  lower  part  of  the  line.  If  the  pointing  is  no  longer  perfect,  the 
micrometer  box  (or  reticle)  must  be  rotated  about  the  axis  of  figure  of  the  telescope  until 
the  wire  (or  line)  is  in  such  a  position  that  this  test  fails  to  discover  any  error. 

To  adjust  the  collimation  proceed  in  the  following  manner:  If  a  transit  micrometer  is  used, 
place  the  micrometer  wire  in  its  mean  position,  as  indicated  by  the  middle  point  of  the  rack  or 
comb  in  the  apparent  upper  (or  lower)  edge  of  the  field,  the  graduated  head  reading  zero. 
Point  on  some  well-defined  distant  object  by  means  of  the  azimuth  screws,  keeping  the  wire 
in  the  position  indicated  above.  Reverse  the  telescope  in  its  wyes  and  again  observe  the  distant 
object.  If  the  wire  again  bisects  the  object,  the  instrument  has  no  error  of  collimation.  If 
upon  reversal  the  wire  does  not  again  bisect  the  object,  then  the  adjustment  is  made  by  bringing 
the  wire  halfway  back  to  the  object  with  the  screw  x,  illustration  No.  5.  Set  on  the  object 
again,  using  the  azimuth  screws,  and  test  the  adjustment  by  a  second  reversal  of  the  telescope, 

If  the  transit  has  a  diaphragm  instead  of  a  transit  micrometer,  the  process  is  very  similar 
to  that  described  above,  though  simpler.  Point  on  some  well-defined  distant  object,  using  the 


16  U.    S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.    14. 

middle  vertical  line  of  the  diaphragm.  Reverse  the  instrument  in  its  wyes  and  again  obseive 
the  same  distant  object.  If  after  reversal  the  wire  covers  the  object  no  adjustment  is 
needed.  If  an  adjustment  is  necessary  it  is  made  by  moving  the  diaphragm  halfway  back  to 
the  object  by  means  of  the  adjusting  screws  which  hold  it  in  place.  A  second  test  should  be 
made  to  show  whether  the  desired  condition  has  been  obtained. 

Wherever  practicable,  the  adjustment  for  collimation  should  be  made  at  sidereal  focus 
on  a  terrestrial  object  at  least  1  mile  distant,  or  on  the  cross  wires  of  a  theodolite  or  collimator 
which  has  previously  been  adjusted  to  sidereal  focus,  set  up  just  in  front  of  the  telescope  of  the 
transit.  If  necessary-  the  lines  of  the  theodolite  are  artificially  illuminated.  Occasionally,  if 
neither  a  distant  object  nor  a  theodolite  is  available  for  making  the  collimation  adjustment, 
a  near  object  may  be  used  for  the  purpose.  In  this  case,  however,  collimation  error  may  exist 
when  the  telescope  is  in  sidereal  focus.  If  such  error  is  not  large,  the  method  of  computations  of 
the  observations  will  eliminate  its  effect  from  the  results.  A  rapid  and  careful  observer  may 
sometimes  be  able  to  make  this  collimation  adjustment  on  a  slow-moving  close  circumpolar 
star.  In  so  doing  he  will  have  to  estimate  the  amount  the  star  moves  while  he  is  reversing  his 
instrument  and  securing  the  second  pointing.  No  attempt  should  be  made  to  adjust  the 
collimation  error  to  zero.  If  it  is  already  less  than  say  0.2  second  of  time  it  should  not  be 
changed,  for  experience  has  shown  that  frequent  adjustment  of  an  instrument  causes  looseness 
in  the  screws  and  the  movable  parts. 

To  test  a  finder  circle  which  is  supposed  to  read  zenith  distances,  point  upon  some  object, 
placing  the  image  of  the  object  midway  between  the  two  horizontal  lines  (guide  lines) ;  bring  the 
bubble  of  the  finder  circle  level  to  the  center  and  read  the  circle.  Next  reverse  the  telescope 
and  point  again  on  the  same  object;  bring  the  bubble  to  the  center  and  read  the  same  finder 
circle  as  before.  The  mean  of  the  two  readings  is  the  true  zenith  distance  of  the  object,  and 
their  half  difference  is  the  index  error  of  the  circle.  The  index  error  may  be  made  zero  by  set- 
ting the  circle  to  read  the  true  zenith  distance,  pointing  on  the  object,  and  bringing  the  vernier 
bubble  to  the  center  with  the  level  adjusting  screw.  At  night  this  adjustment  may  be  made 
by  keeping  a  known  star  between  the  horizontal  lines  as  it  transits  the  meridian.  While  the 
telescope  remains  clamped  in  this  position  set  the  finder  circle  to  read  the  known  zenith  dis- 
tance of  the  star  and  bring  the  bubble  to  the  middle  position  of  the  tube  as  before.  A  quick 
test  when  there  are  two  finder  circles  is  to  set  them  at  the  same  angle  and  see  if  the  bubbles 
come  to  the  center  for  the  same  position  of  the  telescope. 

Adjust  the  transit  micrometer  so  that  it  will  give  20  records  which  are  symmetrical  about 
the  mean  position  of  the  micrometer  wire.  For  a  description  of  this  adjustment  see  page  10. 

The  preceding  adjustments  can  not  always  be  made  in  the  order  named,  as,  for  instance,  when 
a  distant  mark  cannot  be  seen  in  the  meridian,  nor  need  they  all  be  made  at  every  station.  The 
observer  must  examine  and  correct  them  often  enough  to  make  certain  that  the  errors  are 
always  within  allowable  limits. 

The  azimuth  adjustment. — In  the  evening,  before  the  regular  observations  are  commenced, 
it  will  be  necessary  to  put  the  telescope  more  accurately  in  the  meridian.  If  the  chronometer 
correction  is  only  known  approximately,  say  within  one  or  two  minutes,  set  the  telescope  for 
some  bright  star  which  is  about  to  transit  within  10°,  say,  of  the  zenith.  Observe  the  chro- 
nometer time  of  transit  of  the  star.  This  star  being  nearly  in  the  zenith,  its  time  of  transit 
will  be  but  little  affected  by  the  azimuth  error  of  the  instrument.1  The  collimation  and  level 
errors  having  previously  been  made  small  by  adjustment,  the  right  ascension  of  this  star  minus 
its  chronometer  time  of  transit  will  be  a  close  approximation  to  the  chronometer  correction. 
Now  set  the  telescope  for  some  star  of  large  dech'nation  (slow-moving)  which  is  about  to  transit 
well  to  the  northward  of  the  zenith.  Compute  its  chronometer  time  of  transit,  using  the  chro- 
nometer correction  just  found.  As  that  time  approaches  bisect  the  star  with  the  micrometer 

1  To  avoid  waiting  for  stars  close  to  the  zenith  the  chronometer  correction  may  also  be  estimated  closely  by  comparing  observations  of  two  stars 
not  very  distant  from  the  zenith,  one  north  and  one  south,  and  these  at  tte  same  time  will  give  some  idea  of  the  amount  and  direction  of  the  azimuth 
error. 


DETERMINATION   OF   TIME.  17 

wire  in  its  mean  position  or  with  the  middle  vertical  line  of  the  diaphragm  and  keep  it  bisected, 
following  the  motion  of  the  star  in  azimuth  by  the  slow-motion  screws  provided  for  that  pur- 
pose, until  the  chronometer  indicates  that  the  star  is  on  the  meridian. 

The  adjustment  may  be  tested  by  repeating  the  process;  that  is,  by  obtaining  a  closer 
approximation  to  the  chronometer  error  by  observing  another  star  near  the  zenith  and  then 
comparing  the  computed  chronometer  time  of  transit  of  a  slow-moving  northern  star  with 
the  observed  chronometer  time  of  transit.  If  the  star  transits  apparently  too  late,  the  objective 
is  too  far  west  (if  the  star  is  above  the  pole),  and  vice  versa.  The  slow-motion  azimuth  screw 
may  then  be  used  to  reduce  the  azimuth  error.  This  process  of  reducing  the  azimuth  error 
will  be  much  more  rapid  and  certain  if,  instead  of  simply  guessing  at  the  movement  which  must 
be  given  the  azimuth  screw,  one  computes  rouglily  what  fraction  of  a  turn  must  be  given  to  it. 
This  may  be  done  by  computing  the  azimuth  error  of  the  instrument  rouglily  by  the  method 
indicated  on  page  35,  having  previously  determined  the  value  of  one  turn  of  the  screw.1 

If  from  previous  observations  the  chronometer  correction  is  known  within,  say,  five  seconds, 
the  above  process  of  approximation  may  be  commenced  by  using  a  northern  star  at  once,  instead 
of  first  observing  a  zenith  star  as  indicated  above. 

Or,  the  clironometer  correction  being  known  approximately,  and  the  instrument  being  fur- 
nished with  a  screw  or  graduated  arc  with  which  a  small  horizontal  angle  may  be  measured, 
the  first  approximation  to  the  meridian  may  be  made  by  observing  upon  Polaris,  computing  the 
azimuth  approximately  by  use  of  tables  of  azimuth  of  Polaris  at  different  hour  angles  then  by 
means  of  the  screw  or  graduated  arc  swinging  the  instrument  into  the  meridian.  The  tables 
referred  to  are  given  in  Appendix  No.  10  of  the  Report  for  1895,  in  "Principal  Facts  of  the 
Earth's  Magnetism,  etc.,"  (a  publication  of  the  Coast  and  Geodetic  Survey),  or  in  the  Ameri- 
can Ephemeris  and  Nautical  Almanac.  Where  saving  of  time  is  an  important  consideration, 
the  latter  method  has  the  advantage  that  Polaris  may  be  found  in  daylight,  when  the  sun  is 
not  too  high,  by  setting  the  telescope  at  the  computed  altitude  and  moving  it  slowly  in  azi- 
muth near  the  meridian.  It  is  advisable  to  use  a  hack  chronometer  and  the  eye  and  ear 
method  in  making  the  azimuth  adjustments,  the  chronograph  being  unnecessary  for  this  pur- 
pose, even  when  available. 

OBSERVING   LIST. 

The  following  is  an  example  of  the  list  of  stars  selected  for  time  observations  at  stations  of 
a  lower  latitude  than  50°.     The  second  time  set  shown  in  this  list  is  computed  on  page  26,  and 
enters  into  the  longitude  determination  shown  on  page  84.     Each  set  consists  of  two  half  sets 
of  six  stars  each,  selected  hi  accordance  with  the  instructions  shown  on  page  80.     Such  a  list 
prepared  in  easily  legible  figures,  should  be  posted  in  the  observatory. 

1  Some,  of  the  meridian  telescopes  carry  a  small  graduated  arc  on  the  double  base  of  the  frame,  which  may  be  used  for  measuring  the  small  angle 
here  required. 

813C°— 13 2 


18 


Form  250.* 


XI.   S.   COAST  AND  GEODETIC   SUBVEY  SPECIAL  PUBLICATION  NO.   14. 
Star  list  for  Key  West,  Fla. 


</,=24'  33' 


Cata- 
logue 

Star 

Magni- 
tude 

Right  ascension 
a 

Declination 
S 

Zenith  distance 

C 

Star  factors 

Diurnal 
aberration 

K 

A 

C 

B 

h     m     s 

0             / 

O               / 

Bt 

ft  Tauri 

1.8 

5    20    25 

+28    32 

N    3    59 

-0.08 

1.14 

1.14 

-0.02 

At 

£  Aurigae 

5.0 

26     40 

+32    07 

N     7    34 

-0.15 

1.18 

1.17 

-0.02 

B 

t  Orionis 

2.8 

30    53 

-  5    58 

S  30    31 

+0.51 

1.01 

0.87 

-0.02 

B 

o  Aurigae 

5.7 

38    42 

+49    47 

N  25     14 

-0.66 

1.55 

1.40 

-0.03 

B 

£  Leporis 

3.5 

42    44 

-14    51 

S  39    24 

+0.65 

1.04 

0.80 

-0.02 

A 

v  Aurigae 

3.9 

45    03 

+39    07 

N  14    34 

-0.32 

1.29 

1.25 

-0.02 

B 

S  Aurigae 

3.8 

5    51    52 

+54    17 

N  29     44 

-0.85 

1.71 

1.48 

-0.03 

B 

6  Aurigae 

2.7 

53    23 

+37     12 

N  12    39 

-0.28 

1.26 

1.22 

-0.02 

B 

v  Orionis 

4.4 

6    02    16 

+14    47 

S     9    46 

+0.18 

1.04 

1.02 

-0.02 

B 

i)  Geminor. 

3.3 

09    16 

+22    32 

S     2    01 

+0.04 

1.08 

1.08 

-0.02 

B 

8  Monocer. 

4.5 

18    50 

+  4    38 

S  19    55 

+0.34 

1.01 

0.94 

-0.02 

B 

10  Monocer. 

5.0 

23    22 

-  4    42 

S  29    15 

+0.49 

1.01 

0.88 

-0.02 

B 

5  Monocer. 

4.4 

6    35    51 

+  9    59 

S  14    34 

+0.26 

1.02 

0.98 

-0.02 

A 

</>5  Aurigae 

5.5 

40    02 

+43    40 

N  19    07 

-0.45 

1.38 

1.31 

-0.03 

B 

18  Monocer. 

4.7 

43    01 

+  2    31 

S  22    02 

+0.37 

1.01 

0.93 

-0.02 

B 

6  Geminor. 

3.4 

46     40 

+34    04 

N    9    31 

-0.20 

1.21 

1.19 

-0.02 

B 

£    Geminor. 

3.8 

58    36 

+20    42 

S    3    51 

+0.07 

1.07 

1.07 

-0.02 

B 

63  Aurigae 

5.0 

7    05    16 

+39    28 

N  14    55 

-0.34 

1.30 

1.25 

-0.02 

B 

t   Geminor. 

3.8 

7  19    57 

+27    59 

N    3    26 

-0.07 

1.13 

1.13 

-0.02 

B 

/?  Canis  Min. 

2.9 

22    06 

+  8    29 

S  16    04 

4-0.  28 

1.02 

0.97 

-0.02 

B 

a  Canis  Min. 

0.5 

34    26 

+  5    28 

S  19    05 

+0.33 

1.01 

0.95 

-0.02 

B 

/?  Geminor. 

1.1 

39    38 

+28    15 

N    3    42 

-0.08 

1.13 

1.13 

-0.  02 

B 

JT  Geminor. 

5.5 

41    31 

+33    39 

N    9    06 

-0.19 

1.21 

1.18 

-0.02 

A 

<j>  Geminor. 

5.0 

47    48 

+27    00 

N    2    27 

-0.05 

1.12 

1.12 

-0.02 

*  Form  25fi,  known  as  "Coast  and  Geodetic  Survey,  Longitude  Record  and  Computation,"  is  a  book  containing  all  the  different  forms  used 
in  observing  and  computing,  time  and  longitude,  except  form  34  shown  on  p.  20. 
fBerliner  Astronomisches  Jahrbuch. 
t  American  Ephemeris  and  Nautical  Almanac. 

DIRECTIONS   FOR  OBSERVING. 

Everything  being  in  readiness  and  the  instrument  completely  adjusted  set  the  tele- 
scope for  the  first  star.  It  is  not  advisable  to  use  the  horizontal  axis  clamp  during  obser- 
vations, for  its  action  may  have  a  slight  tendency  to  raise  one  end  of  the  axis.  See  to  it,  loading 
one  end  if  necessary,  that  the  center  of  gravity  of  the  telescope  is  at  its  horizontal  axis,  and  then 
depend  upon  the  friction  at  the  pivots  to  keep  the  telescope  in  whatever  position  it  is  placed. 
Watch  the  chronometer 1  so  as  to  know  when  to  expect  the  star  to  appear  in  the  field  of  view  of  the 
telescope.  When  the  star  enters  the  field,  bring  it  between  the  horizontal  lines  of  the  diaphragm, 
if  it  is  not  already  there,  by  tapping  the  telescope  lightly. 

If  a  transit  micrometer  is  used  the  process  of  observing  consists  simply  in  bisecting  the  star's 
image  with  the  micrometer  wire  soon  after  it  appears  and  in  keeping  it  bisected  as  it  moves 
across  the  field  of  the  telescope.  The  record  is  made  automatically  by  the  contact  of  a  spring 
with  certain  metal  strips  on  the  micrometer  head.  A  cut-out  device  allows  only  10  such  con- 
tacts on  either  side  of  the  moan  position  of  the  micrometer  wire  to  register  on  the  chronograph. 
The  observer  learns  by  experience  at  what  part  of  the  field  the  wire  begins  to  register  and  he 
should  endeavor  to  keep  the  star  bisected  several  seconds  before  it  reaches  that  point.  Similarly, 
he  knows  when  the  record  is  complete  and  he  can  cease  observing  a  particular  star,  and  set  for 
the  next  one  on  his  observing  list. 

If  an  instrument  with  a  diaphragm  is  being  used  in  connection  with  a  chronograph,  the 
process  of  observing  the  transit  of  a  star  across  a  line  of  the  diaphragm  consists  in  waiting, 
observing  key  in  hand,  until  the  instant  when  the  star  is  apparently  bisected  by  the  line  and 
then  pressing  the  key  as  soon  as  possible  thereafter.  The  time  record  thus  made  on  the  chrono- 

i  When  achronograph  is  being  used,  it  is  customary  to  keep  the  chronometer  which  is  connected  with  the  chronograph  protected  as  carefully  as 
possible  from  rapid  changes  of  temperature  and  from  jars.  During  the  observations  it  is  not  usually  removed  from  its  protecting  box,  but  instead 
an  extra  chronometer  (sometimes  called  a  hack  chronometer)  is  used  at  the  instrument. 


DETERMINATION   OF   TIME.  19 

graph  will  always  follow  the  event  by  a  time  interval,  known  as  personal  equation,  which 
depends  mainly  on  the  rapidity  of  the  action  of  the  nerves  and  brain  of  the  observer. 

It  may  occur  to  a  new  observer  to  attempt  to  make  this  time  interval  zero  by  anticipating 
the  bisection  of  the  star's  image,  and  this  he  may  succeed  in  doing.  He  may  even  make  the 
personal  equation  negative.  The  accumulated  experience  of  many  observers,  however,  is  that 
it  is  better  to  observe  in  the  manner  first  indicated  and  have  a  large  and  constant  personal 
equation,  rather  than  to  reduce  this  personal  equation  to  a  small  but  at  the  same  tune  rather 
variable  quantity.  The  method  of  observing  with  a  transit  micrometer  practically  eliminates 
the  personal  equation  from  the  tune  observations.  In  other  methods  it  may  be  eliminated 
from  the  results  by  special  observations,  or  by  programs  of  observing  especially  devised  for 
that  purpose.  (See  p.  91.) 

At  about  the  middle  of  the  observations  which  are  to  constitute  a  set  the  telescope  should 
be  reversed,  so  that  the  effects  of  the  error  of  collimation  and  inequality  of  pivots  upon  the 
apparent  times  of  transit  may  be  reversed  in  sign.  Three  or  four  readings  of  the  striding  level, 
in  each  of  its  positions  (direct  and  reversed)  should  be  taken  during  each  half  set.  To  eliminate, 
in  part  at  least,  the  effects  of  irregularities  in  the  figure  of  the  pivots  upon  the  determination  of 
the  inclination  of  the  axis,  it  is  desirable  to  take  the  level  readings  with  the  telescope  inclined 
at  the  various  practicable  angles  at  which  stars  are  observed,  and  to  make  half  of  them  with  the 
objective  to  the  northward  and  half  with  the  objective  southward.  Great  care  should  be 
taken  to  avoid  unequal  heating  of  the  two  ends  of  the  striding  level.  The  level  readings  may 
be  checked  and  possible  errors  often  detected  by  the  fact  that  the  bubble  length  should  be 
constant  except  for  the  effect  of  change  of  temperature  (the  bubble  shortens  with  rise  of  tem- 
perature) and  in  observing  and  computing  this  should  be  kept  in  mind.  A  very  short  length 
of  bubble  should  not  be  used  on  account  of  increased  tendency  to  stick,  and  extreme  length 
should  be  avoided  because  of  danger  of  running  off  the  graduation.  In  using  the  striding  level 
it  is  important  that  the  bubble  be  given  tune  to  come  to  rest  before  reading. 

The  only  difference  between  the  eye  and  ear  method  of  observing  time  and  the  chronograph 
and  key  method  just  described  is  in  the  process  of  observing  and  recording  the  times  of  transit 
of  the  star  image  across  the  separate  lines  of  the  diaphragm. 

Before  using  the  eye  and  ear  method  the  observer  must  first  learn  to  pick  up  the  beat  of  a 
chronometer  and  to  carry  it  even  while  paying  attention  to  other  matters.  To  pick  up  the 
beat  of  a  chronometer,  first  look  at  some  second's  mark  two  or  more  seconds  ahead  of  the  second 
hand.  Fix  the  number  of  that  second  in  mind  as  the  second  hand  approaches  it.  Name  it 
exactly  with  the  tick  at  which  the  second  hand  reaches  it.  Then,  keeping  the  rhythm  of  the 
chronometer  beat,  count  the  seconds  and  half  seconds  (aloud,  in  a  whisper,  or  mentally),  always 
keeping  the  count  exactly  with  the  tick  of  the  chronometer.  In  counting  it  will  be  found  easier 
to  keep  the  rhythm  if  the  names  of  the  numerals  are  elided  in  such  a  way  as  to  leave  but  a 
single  staccato  syllable  in  each.  The  half -second  beat  should  be  marked  by  the  word  "half," 
thus — one,  half,  two,  half,  three  .  .  .  twenty,  half,  twenty-one,  half,  twenty-too  .  .  .  and  so 
on.1  With  practice,  an  observer  can  carry  the  count  of  the  beat  for  an  indefinite  period 
without  looking  at  the  chronometer  face  if  he  can  hear  the  tick.  If  he  becomes  expert,  he  will 
even  be  able  to  carry  the  count  for  a  half  minute  or  more  during  which  he  has  not  even  heard 
the  tick.  The  chronometer  should,  of  course,  be  placed  where  it  can  be  seen  and  heard  by  the 
observer  with  as  little  effort  as  possible. 

To  observe  the  time  of  transit  of  a  star  across  a  given  line  the  observer  first  picks  up  the 
beat  of  the  chronometer  as  the  star  approaches  the  line.  At  the  last  tick  of  the  chronometer 
occurring  before  the  transit  he  notes  mentally  the  number  of  the  tick,  and  also  carefully  observes 
the  apparent  distance  of  the  star  from  the  line.  At  the  next  tick  the  star  is  on  the  other  side 
of  the  line  and  the  observer  notes  again  the  apparent  distance  of  the  star  from  the  line.  By  a 
mental  comparison  of  these  two  distances  he  estimates  fifths  of  the  time  interval  between  the  two 
ticks  of  the  chronometer  and  obtains  his  estimate  of  the  time  of  transit  to  the  nearest  tenth  of 
a  second.  Though  the  mental  processes  involved  may  seem  difficult  at  first,  practice  soon  makes 
them  easy.  An  experienced  observer  using  this  process  is  able  to  estimate  the  tune  of  transit 

i  Another  method  often  used  is  to  count  only  to  10  (thus  using  only  words  of  one  syllable)  and  to  glance  at  the  chronometer  alter  the  obser- 
vation to  show  the  position  in  the  minute. 


20 


U.   S.   COAST   AND  GEODETIC  SUBVEY   SPECIAL  PUBLICATION   NO.   14. 


of  a  star's  image  across  a  line  of  the  diaphragm  with  a  probable  error  of  about  ±0s.l.  It  is 
conducive  to  accuracy  for  the  observer  to  acquire  the  habit  of  deciding  definitely,  without 
hesitation,  upon  the  second  and  tenth  as  soon  as  the  event  is  complete.  Hesitation  in  this 
matter  is  likely  to  cause  inaccuracy. 

EXAMPLE  OF  RECORD  AND  PART  OF  THE  COMPUTATIONS. 

There  are  shown  on  pages  18,  20-22  examples  of  the  list  of  stars  and  the  original  transit  level 
readings  made  in  the  observatory  at  the  time  of  the  observations,  a  set  of  time  observations 
as  read  from  the  chronograph  sheet,  and  the  computation  of  a  —  t  (right  ascension  minus  the 
chronometer  time  of  transit)  for  each  star.  The  computation  of  AT  (the  mean  correction  to 
the  chronometer)  is  shown  on  page  26.  These  computations  are  for  the  second  set  of  stars 
given  on  page  18. 

These  observations  were  made  under  the  General  Instructions  for  Longitude  Determina- 
tions with  the  Transit-Micrometer,  which  are  given  on  page  79  of  this  publication. 


Form  34. 


Longitude  record. 

[Station,  Key  West.    Date,  Feb.  14, 1907.    Instrument,  Transit  No.  2.    Observer,  J.  S.  Hill.) 


Set  I 

Set  II 

Stars 

Levels 
W                     E 

Stars 

Levels 
W                     E 

d      N        d 

d      N        d 

Clamp  or  band,  W 

17.  7        58.  8 

Clamp  or  band,  W 

62.  0        20.  0 

ft  Tauri 

60.  1        19.  0 

S    Monocer. 

17.  7        59.  5 

%  Aurigae 

<f>  5  Aurigae 

i  Orionis 

S 

18  Monocer. 

S 

o  Aurigae 

17.  7        58.  8 

6  Geminor. 

61.  2        19.  4 

v  Aurigae 

61.  2        20.  0 

£  Geminor. 

17.  7        59.  6 

63  Aurigae 

N 

N 

17.  5        58.  9 

61.5        19.5 

60.7        19.3 

17.  7        59.  7 

S 

17.6        59.0 

61.  7        20.  2 

N 

N 

Clamp  or  band,  E 

17.  0        58.  7 

Clamp  or  band,  E 

16.  8        58.  9 

S  Aurigae 

61.3        19.7 

i  Geminor. 

61.  6        19.  5 

6  Aurigae 

ft  Canis  Min. 

j]  Geminor. 

S 

a  Canis  Min. 

S 

8  Monocer. 

17.2        59.0 

ft  Geminor. 

17.4        59.7 

10  Monocer. 

61.  9        20.  0 

n  Geminor. 

62.  1        19.  7 

<j>  Geminor. 

N 

N 

16.  8        58.  7 

17.  0        59.  4 

61.3        19.4 

62.  0        19.  5 

S 

16.  9        59.  4 

62.  3        19.  9 

1  div.  of  level  scale  —  2". 322.       Chronometer  1824. 

Pivot  inequality  =  0.000. 

Remarks:  Cable  was  used  direct,  without  repeaters,  between  Miama  and  Key  West. 


DETERMINATION  OF   TIME. 


21 


While  the  following  method  of  computing  was  devised  for  observations  with  the  transit 
micrometer,  it  is  not  limited  in  its  use  to  such  observations.  The  star  list  for  which  observa- 
tions and  computations  are  shown  on  the  following  pages  could  have  been  observed  with  a 
key  and  the  computation  made  in  the  same  manner  as  the  one  which  foUows.  The  only  differ- 
ence is  that  had  the  observations  been  made  with  a  key  not  so  many  records  would  have  been 
obtained  and  the  observations  would  have  been  subject  to  a  large  observation  error,  called 
personal  equation.  (See  p.  90.) 

Explanation  of  the  formulae  and  methods  used  hi  this  computation  follows  the  examples 
ol  the  record  and  computation. 

Form  256.* 

[Station,  Key  West.    Date,  Feb.  14, 1907.    Instrument,  transit  No.  2,  with  transit  micrometer.    Observer,  J.  S.  Hill.    Recorder,  J.  S.  Hill.    Cnro- 

nometer,  Sidereal  1824.] 


Star:  S.  Monoccr. 

ifi'  Aurigae 

18  Monocer. 

£  Geminor. 

C  Geminor. 

63  Aurigae 

Clamp:  W 

W 

W 

W 

VV 

W 

Lev 

el: 

W 

E 

W            E 

W 

E 

d 

d 

d             d 

d 

d 

N62.0 

20.0 

S61.2          19.4 

N61.5 

19.5 

17.7 

59.5 

17.  7         59.  6 

17.7 

59.7 

+44.3 

-39.5 

+43.5       -40.2 

+43.8 

-40.2 

+4.S 

+3.3 

+3.f 

i 

Computatior 

of  level  constant:  Me 

anN+4.20 

S+3.30 



s 

+  3.  75X0.039-  +0.140=  bw 

h  m 

h  m 

h  m 

h  m 

h  m 

K  m 

6    35 

6    39 

6    42 

6    46 

6    58 

7    04 

s 

s 

Sums 

s          s 

Sums 

s         s        Sums 

s          s 

Sums 

s          s 

Sums 

s         s       Sums 

32.0 

41.4 

73.4 

41.3     54.0 

95.3 

41.5     50.5        92.0 

19.  5     30.  4 

49.9 

16.2     26.0 

42.2 

55.3     67.0       122.3 

32.4 

41.1 

.5 

41.8     53.5 

.3 

41.9     50.2            .1 

20.  0     30.  1 

50.1 

16.5     25.5 

2.0 

55.  6     66.  5            .1 

33.1 

40.4 

.5 

42.8     52.6 

.4 

42.  5     49.  7            .2 

20.  6     29.  4 

.0 

17.  2     24.  8 

2.0 

56.  4     65.  8            .2 

33.6 

39.8 

.4 

43.5     51.9 

.4 

43.  1     49.  1            .2 

21.3     28.7 

.0 

17.7     24.3 

2.0 

57.  1     65.  1            .2 

33.9 

39.5 

.4 

43.9     51.4 

.3 

43.  3     48.  8            .1 

21.  7     28.  3 

.0 

18.  0     23.  9 

1.9 

57.  5     64.  6            .1 

34.6 

38.8 

.4 

44.7     50.6 

.3 

44.  0     48.  1            .1 

22.  3     27.  6 

49.9 

18.  8     23.  1 

1.9 

58.  4     63.  9           .3 

35.0 

38.5 

.5 

45.3     50.3 

.6 

44.  3     47.  9            .2 

22.  8     27.  1 

9.9 

19.1     22.9 

2.0 

58.  8     63.  4             .2 

35.6 

37.9 

.5 

46.0     49.3 

.3 

44.8     47.3            .1 

23.6     26.4 

50.0 

19.  8     22.  3 

2.1 

59.  5     62.  6            .1 

36.1 

37.4 

.5 

46.9     48.5 

.4 

45.  4     46.  6            .0 

24.  3     25.  7 

.0 

20.  5     21.  6 

2.1 

60.3     61.9            .2 

36.4 

37.1 

.5 

47.  2     48.  1 

.3 

45.  7     46.  3            .0 

24.  6     25.  4 

.0 

20.7     21.4 

2.1 

60.7     61.5            .2 

Sum    734.  6 

Sum    953.6 

Sum    921.  0 

Sum    499.  8 

Sum    420.  3 

Sum  1221.9 

Mean 

36.73 

47.68 

46.05 

24.99 

21.02 

01.10 

Rt 

K 

-     .02 

-    .03 

-    .02 

-    .02 

-    .02 

-    .02 

Bb 

+     .14 

+    .19 

+    .14 

+    .17 

+    .16 

+    .18 

t 

6    35 

36.85 

6    39 

47.84 

6    42       46.17 

6    46 

25.14 

6    58 

21.16 

7    05       01.26 

a. 

6    35 

51.85 

6    40 

02.92 

6    43        01.21 

6    46 

40.17 

6    58 

36.16 

7    05        15.28 

(a-«) 

+  15.00 

+  15.08 

+  15.04 

+  15.03 

+  15.00 

+15.02 

*  See  note  below  table  on  p.  18. 

t  K,  correction  for  rate,  is  negligible  in  this  time  set. 


22 


U.   S.   COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 


Form  256.* 

[Station,  Key  West.    Date,  Feb.  14, 1907.    Instrument,  transit  No.  2,  with  transit  micrometer.    Observer,  J.  S.  Hill.    Recorder,  J.  S.  Hill.    Chro- 
nometer, Sidereal  1824.] 


Star 

:  t  Geminor. 

p  Canis  Min. 

tt  Canis  Min. 

f  Geminor. 

?r  Geminor. 

$  Geminor 

Clamp:  E 

E 

E 

E 

E 

E 

Level: 

\V 

E 

W             E 

W 

E 

W             E 

d 

d 

d             d 

d 

d 

d             d 

N 

16.8 

58.9 

S   17.4         59.7 

N  17.  0 

59.4 

S  16.  9         59.  4 

61.6 

19.5 

62.  1          19.  7 

62.0 

19.5 

62.  3           19.  9 

+44.8 

-39.4 

+44.7       -40.0 

+45.0 

-39.9 

+4.5.4       —39.5 

+5.4 

+4.7 

+5.1 

+5.9 

d 

Computation  of  level  constant:  Mean  N  +5.  25 

S+5.30 

+5.  28X0.039=  +0.206=  &E 

ft      771 

h    m 

h    m 

h    m 

h    m 

h    in 

7     19 

7    21 

1    34 

7    3!) 

7    41 

7    47 

s 

s 

Sums 

s          5       Sums 

a          s        Sums 

s         s 

Sums 

s          s       Sums 

s          s        Sums 

37.8 

48.3 

86.1 

47.9     57.1       105.0 

07.5     16.7        24.2 

18.  5     28.  8 

47.3 

11.3     22.3         33.6 

29.  5     39.  6        69.  1 

38.3 

47.9 

.2 

48.  2     56.  8          5.  0 

07.  8      16.  4             .2 

18.  8     28.  5 

.3 

11.6     21.9            .5 

29.8     39.4            .2 

38.9 

47.3 

.2 

48.  7     56.  1          4.  8 

08.  4     15.  7            .1 

19.5     27.7 

.2 

12.5     21.1            .6 

30.3     38.5         68.8 

39.6 

46.5 

.1 

49.3     55.5          4.8 

09.  0     15.  1            .1 

20.1     27.0 

.1 

13.  2     20.  4            .6 

31.  0     37.  8            .8 

39.9 

46.3 

.2 

49.  7     55.  2          4.  9 

09.  2     14.  8            .0 

20.  5     26.  8 

.3 

13.  6     20.  1            .7 

31.3     37.5            .8 

40.7 

45.6 

.3 

50.  2     54.  6          4.  8 

09.9     14.2            .1 

21.  2     26.  1 

.3 

14.  3      19.  4             .7 

32.  0     36.  8            .8 

41.0 

45.1 

.1 

50.  6     54.  4          5.  0 

10.  2     13.  9            .1 

21.  6     25.  7 

.3 

14.  7      19.  0             .7 

32.  3     36.  5            .8 

41.7 

44.6 

.3 

51.  1     53.  7          4.  8 

10.8     13.3            .1 

22.3     25.0 

.3 

15.4     18.3            .7 

33.  1     35.  9         69.  0 

42.5 

43.8 

.3 

51.  8     53.  0          4.  8 

11.  4     12.  6            .0 

23.1     24.3 

.4 

16.1      17.5             .6 

33.  8     35.  1         68.  9 

42.8 

43.4 

.2 

52.  1     52.  7          4.  8 

11.7     12.3            .0 

23.3     24.1 

.4 

16.3      17.2             .5 

34.  1     34.  8            .9 

Sum    862.  0 

Sum  1048.7 

Sum    240.  9 

Sum 

472.9 

Sum    336.  2 

Sum    689.  1 

Mean 

43.10 

52.44 

12.04 

23.64 

16.81 

34.46 

Rt 

X 

-     .02 

-     .02 

-     .02 

-     .02 

-     .02 

-     .02 

Bb 

+     .23 

+     .20 

+     .20 

+     .23 

+     .24 

+     .23 

t      7 

19 

43.31 

7    21        52.63 

7    34        12.22 

7    39 

23.85 

7     41         17.03 

7    47        34.67 

a    7 

19 

57.74 

7    22        07.  08 

7    34        26.67 

7    39 

38.26 

7     41        31.45 

7    47        49.14 

(a—  t 

) 

+  14.43 

+  14.45 

+  14.45 

+  14.41 

+  14.42 

+  14.47 

*  Eee  note  below  table  on  p.  18. 

t  R,  correction  lor  rate,  is  negligible  in  this  time  set. 

CORRECTION   FOR  INCLINATION   OF  AXIS. 

If  the  horizontal  axis  of  the  telescope  is  slightly  inclined  to  the  horizon  and  the  telescope 
is  otherwise  in  perfect  adjustment,  the  line  of  collimation  will,  when  the  telescope  is  rotated 
about  its  horizontal  axis,  describe  a  plane  which  passes  through  the  north  and  south  points  of 
the  horizon  and  makes  an  angle  with  the  meridian  plane  equal  to  the  inclination  of  the  axis  to 
the  horizon.  If  the  eastern  end  of  the  axis  is  too  high,  the  transits  of  all  the  stars  above  the 
pole  (apparently  moving  westward)  will  be  observed  too  late,  and  the  transits  of  all  subpolars 
will  be  observed  too  early,  and  it  is  therefore  necessary  to  correct  the  observed  times  of  transit 
by  means  of  the  readings  of  the  striding  level,  taking  into  account  the  inequality  of  the  pivots, 
if  appreciable. 

Let  w  and  e  be  the  readings  of  the  west  and  east  ends,  respectively,  of  the  bubble  of  the 
striding  level  for  a  given  position  of  the  telescope  axis.  Let  w'  and  e,'  be  the  corresponding  west 
and  east  readings  after  the  level  is  reversed,  the  telescope  axis  remaining  as  it  was.  Let  d  be 
the  value  of  a  division  of  the  level  in  seconds  of  arc.  Then  for  /3,  the  apparent  inclination  of  the 


DETERMINATION    OF    TIME.  23 

telescope  axis  expressed  in  seconds  of  time,  we  may  write,  if  the  level  divisions  are  numbered 
in  both  directions  from  the  middle  : 

f)  -  (e  +  e1)  }  ~  =  [  (w  +  wf)  -(«  +  «')  1  4 

)   1O         I  J  DU 

in  whicli  ^  is  a  constant  for  the  level,  -r-=  being  the  value  of  one  division  of  the  level  in  seconds 
ou  lo 

of  time. 

If  the  level  divisions  are  numbered  continuously  from  one  end  of  the  level  to  the  other  the 
above  formula  takes  the  form 

/?=     (w-wf)  +  («-«')      L 


in  whicli  the  primed  letters  refer  to  that  position  of  the  level  in  which  the  zero  end  of  the  tube 
is  to  the  west.1 

Inequality  of  pivots.  —  The  level  readings  give  a  determination  of  the  inclination  of  the  line 
joining  the  points  of  the  two  pivots,  which  are  midway  between  the  lines  of  contact  of  the  pivots 
and  the  wyes  of  the  level,  but  do  not  give  the  required  inclination  of  the  axis  of  rotation  of  the 
telescope  (which  is  the  line  joining  the  centers  of  the  two  pivots)  unless  the  pivots  are  of  the  same 
size.  Let  p,  the  pivot  inequality,  be  the  angle,  expressed  in  seconds  of  time,  between  the  line 
joining  the  centers  of  the  pivots  and  the  line  whose  inclination  is  determined  by  the  level  readings, 
and  let  this  angle  be  called  positive  if  the  pivot  nearest  the  designating  mark  (band,  clamp,  or 
illumination)  is  the  smaller. 

Then 

and  bE  =  3e-     2 


in  which  b  is  the  required  inclination  of  the  axis  of  rotation  of  the  telescope.  The  subscripts 
indicate  the  position,  to  the  westward  or  to  the  eastward,  of  the  bright  band,  the  clamp,  or  the 
illumination,  or  whatever  mark  is  used  to  distinguish  between  the  two  positions  of  the  telescope 
axis.  The  pivot  inequality,  p,  is  ordinarily  derived  from  a  special  series  of  observations  taken 
for  that  purpose.  For  an  example  of  such  a  series,  with  the  corresponding  formula  and  com- 
putation, see  page  44. 

The  correction  to  the  observed  time  of  transit  of  any  star  for  inclination  is 

b  cos  £  sec  d  =  bB, 

in  which  d  is  the  declination  of  the  star  and  £  is  its  zenith  distance  (  =</>  —  S  for  all  stars  above 
the  pole,  and  =<j>  +  d—  180°  for  subpolar  stars)  .  The  factor  B  =  cos  £  sec  3  is  tabulated  on  pages 
62-77,  but  is  much  more  easily  obtained  with  the  graphical  device  shown  in  illustration  No.  9 
and  explained  on  page  61.  It  is  positive  for  stars  above  the  pole  and  negative  for  subpolars. 

It  is  the  present  practice  in  this  Survey  to  assume  that  b,  the  inclination,  is  constant  for 
each  half  set,  and  it  is  computed  in  the  following  manner:  Within  each  half  set  the  mean  of  the 
observed  values  of  j)  with  objective  northward  is  first  derived,  then  the  corresponding  mean 
with  objective  southward,  and  finally  the  mean  of  these  two  means  is  taken  as  the  /?  for  the 
half  set. 

The  value  of  B  for  each  star,  as  taken  from  either  the  table  on  pages  62-77  or  the  graphical 
device  shown  in  illustration  No.  9,  is  given  in  the  observing  list  on  page  18. 

i  As  w  is  always  greater  than  w'  and  «  is  always  less  than  t',  the  sign  of  the  west  difference  is  always  +  and  of  the  east  difference  is  always  —  , 
so  that  when  the  differences  are  taken  vertically,  the  resulting  sign  of  the  level  correction  will  at  once  be  apparent,  as  shown  in  the  following 
example: 

West  East 

d  d 

62.  0  20.  0 

17.7  S9.S 

+44.3  -39.5 

+4.8 

s  These  formulae  are  exact  only  in  case  the  angle  of  the  level  wyes  is  the  same  as  the  angle  of  the  supporting  wyes. 


24 


U.   S.   COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 


INCOMPLETE  TRANSITS  WITH  TRANSIT-MICROMETER. 

If  the  transit  of  a  star  observed  with  the  transit-micrometer  is  incomplete,  only  the  obser- 
vations which  are  symmetrical  with  regard  to  the  mean  position  of  the  micrometer  wire  are 
used  and  those  for  wliich  the  symmetrical  observations  are  lacking  are  rejected.  (See  General 
Instructions  for  Longitude  Determinations,  p.  79.)  Incomplete  transits  by  other  methods  of 
observing  are  utilized  by  a  method  of  reduction  shown  on  page  32. 

CORRECTION   FOR  RATE. 

If  the  chronometer  rate  is  not  zero,  the  chronometer  correction  changes  during  the  progress 
of  the  time  set.  To  reduce  each  observed  time  of  transit  across  the  mean  line  to  what  it  would 
have  been  had  the  rate  been  zero  (and  the  correction  equal  to  that  which  actually  existed  at 
the  mean  epoch  of  the  set)  apply  the  following  correction : 

R=(t-T0)rh 

in  which  t  is  the  chronometer  time  of  transit  of  a  star,  T0  is  the  mean  epoch  of  the  time  set,  that 
is,  the  mean  of  ah1  the  chronometer  times  of  transit,  and  rh  is  the  hourly  rate  of  the  chronometer 
on  sidereal  time,  +  when  losing  and  --  when  gaining.  The  quantity  (t—  T0)  is  expressed  in 
hours.  The  above  is  the  correction  as  applied  to  the  observed  time  of  transit  of  the  star;  applied 
to  a  —  t,  the  sign  is  reversed. 

The  correction  for  rate  may  be  looked  upon  as  a  refinement  which  is  not  always  essential. 
If  a  time  set  has  perfect  symmetry  of  arrangement,  the  effect  of  introducing  a  rate  correction 
into  the  computation  will  be  shown  only  in  the  residuals,  as  it  will  have  no  effect  on  the  com- 
puted clock  correction.  If  the  daily  rate  of  the  chronometer  is  less  than  five  seconds,  it  can  be 
ignored  in  the  computation  of  all  time  sets  except  those  in  which  one  of  the  half  sets  contains 
many  more  or  less  stars  than  the  other,  or  in  which  one  of  the  half  sets  extends  over  a  very 
much  longer  period  of  time  than  the  other.  In  all  cases  where  the  rate  is  greater  than  five  seconds 
per  day  it  should  be  considered,  and  it  should  be  omitted  only  after  a  preliminary  test  shows  its 
effect  on  the  chronometer  correction  to  be  negligible. 

CORRECTION  FOR  DIURNAL  ABERRATION. 

The  effect  of  the  annual  aberration  due  to  the  motion  of  the  earth  in  its  orbit  is  taken  into 
account  in  computing  apparent  star  places  and  need  not  be  considered  here. 

The  correction  for  diurnal  aberration  to  be  applied  to  an  observed  tune  of  transit  across 
the  meridian  is 

K=08.021  cos  <£  sec  § 

This  correction  may  be  obtained  easily  by  the  graphical  device  shown  in  illustration  No.  9 
and  described  on  page  61,  but  it  is  also  given  in  the  following  table.  It  is  minus  for  all  stars 
observed  at  upper  culmination  and  plus  for  stars  observed  at  lower  culmination. 

Table  of  diurnal  aberration  (K). 


Latitude 

Declination-,? 

-* 

0" 

10° 

20° 

30° 

40° 

50° 

60° 

70° 

75° 

80° 

85° 

S 

s 

S 

S 

S 

S 

S 

S 

S 

* 

* 

0° 

0.02 

0.02 

0.02 

0.02 

0.03 

0.03 

0.04 

0.06 

0.08 

0.12 

0.24 

10° 

.02 

.02 

.02 

.02 

.03 

.03 

.04 

.06 

.08 

.12 

.24 

20° 

.02 

.02 

.02 

.02 

.03 

03 

.04 

.06 

.08 

.11 

.23 

30° 

.02 

.02 

.02 

.02 

.02 

.03 

.04 

.05 

.07 

.10 

.21 

40° 

.02 

.02 

.02 

.02 

.02 

.03 

.03 

.05 

.06 

.09 

.18 

50° 

.01 

.01 

.01 

.02 

.02 

.02 

.03 

.04 

.05 

.08 

.15 

60° 

.01 

.01 

.01 

.01 

.01 

.02 

.02 

.03 

.04 

.06 

.12 

70° 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.02 

.03 

.04 

.08 

80° 

.00 

.00 

.00 

.00 

.00 

.01 

.01 

.01 

.01 

.02 

.04 

DETERMINATION   OF   TIME.  25 

DERIVATION  OF  («-<)• 

The  correction  for  diurnal  aberration,  inclination  of  axis,  and  rate  (if  considered)  being 
applied  to  the  observed  time  of  transit  across  the  mean  position  of  the  micrometer  wire  (or 
mean  line  of  the  diaphragm)  as  shown  in  the  computation  on  pages  21-22,  the  result  ist,  an  approxi- 
mate time  of  transit  across  the  meridian.  The  apparent  right  ascension  at  the  time  of  observa- 
tion is  taken  from  some  star  catalogue,  giving  apparent  places,  such  as  the  American  Ephemeris 
and  Nautical  Almanac  or  the  Berliner  Astronomisches  Jahrbuch  (pieferably  the  former)  The 
difference  between  t  and  the  right  ascension,  a,  of  the  star  at  the  time  of  observation,  is  (ac  —  t). 
an  approximate  correction  to  the  chronometer  time. 

In  taking  right  ascensions  from  the  star  catalogue  it  is  necessary  to  interpolate  for  the 
longitude  of  the  observer,  and  to  consider  second  differences  when  they  affect  the  result  by  as 
much  as  a  hundred tli  of  a  second. 

THE  COLLIMATION  CORRECTION. 

If  the  instrument  is  otherwise  in  perfect  adjustment,  but  has  a  small  error  in  collimation, 
the  micrometer  wire  in  its  mean  position  (or  the  mean  line  of  the  diaphragm)  will  describe  a 
small  circle  parallel  to  the  meridian  and  at  an  angular  distance,  the  error  of  collimation,  from  it, 
when  the  telescope  is  rotated  about  its  horizontal  axis. 

The  collimation  correction  =  c  sec  o  =  Cc, 

in  which  c  is  the  angle,  expressed  in  seconds  of  time,  between  the  line  of  sight  defined  by  the 
micrometer  wire  when  in  its  mean  position  (or  by  the  mean  line  of  the  diaphragm)  and  a  plane 
perpendicular  to  the  horizontal  axis  of  the  telescope.  In  other  words,  c  is  the  angle  between  the 
line  of  collimation  and  the  collimation  axis.  (See  p.  13.)  It  is  considered  positive  for  a  given 
telescope  if  the  line  of  sight  is  too  far  east  (and  stars  at  upper  culmination  are  therefore  observed 
too  soon)  when  the  illumination  (or  bright  band)  is  to  the  westward.  This  convention  of  sign 
is  purely  arbitrary,  however,  c  is  derived  from  the  time  computations  by  one  of  the  processes 
shown  on  pages  26,  34,  and  42. 

The  factor  C  is  written  for  sec  d  and  is  tabulated  on  pages  62-77.  It  is  more  easily  obtained 
from  the  graphical  device  shown  in  illustration  No.  9  and  described  on  page  61.  For  observa- 
tions made  with  illumination  (or  band)  to  the  westward  C  is  to  be  considered  positive  for  stars 
at  upper  culmination  and  negative  for  stars  at  lower  culmination.  The  signs  are  reversed  with 
illumination  (or  band)  east. 

THE  AZIMUTH  CORRECTION. 

If  the  instrument  is  otherwise  in  adjustment,  but  has  a  small  error  in  azimuth,  the  microme- 
ter wire  in  its  mean  position  (or  the  mean  line  of  the  diaphragm)  will  describe  a  vertical  circle 
on  the  celestial  sphere  at  an  angle  with  the  meridian.  The  correction  in  seconds  to  an  observed 
time  of  transit  for  this  azimuth  error  is, 

Azimuth  correction  =  a  sin  £  sec  d  =  Aa, 

in  which  a  is  the  angle  expressed  in  seconds  of  time  between  the  meridian  and  the  vertical  circle 
described  by  the  mean  position  of  the  micrometer  wire.1  It  is  considered  positive  when  the 
collimation  axis  is  too  far  to  the  east  with  the  telescope  pointed  south. 

For  convenience  A  is  written  for  sin  £  sec  3  and  will  be  found  tabulated  on  pages  62-77. 
It  can  be  more  easily  obtained  with  the  graphical  device  shown  in  illustration  No.  9  and  described 
on  page  61.  The  factor  A  is  considered  positive  for  all  stars  except  those  between  the  zenith 
and  the  pole. 


'  In  practice  there  always  exists  an  error  of  collimation,  so  in  general  a  is  tha  angle  between  the  meridian  and  the  axis  of  collimation. 


26 


TJ.   S.   COAST   AND   GEODETIC   SUBVEY   SPECIAL   PUBLICATION   NO.   14. 


a  is  derived  from  the  observations  by  one  of  the  processes  shown  on  pages  26,  34,  39,  and 
42,  attention  being  paid  to  sign  as  indicated  above. 

COMPUTATION  OF  AT,  c,  AND  a  WITHOUT  LEAST  SQUARES. 

The  following  method  of  computation  was  devised  shortly  after  the  tune  (1905)  the  transit- 
micrometer  was  adopted  by  this  survey  for  use  on  longitude  work  and  it  is  used  both  in  the  field 
and  in  the  office  for  the  final  computation  of  ah1  tune  observations  made  with  the  transit  microme- 
ter at  stations  in  latitude  less  than  50°.  In  all  latitudes  greater  than  50°  the  least-square 
solution  is  used  in  obtaining  the  final  results.  There  is  also  a  somewhat  different  method  of 
computation  (shown  on  p.  34)  used  when  the  stars  of  a  time  set  consist  of  four  time  stars  and 
one  azimuth  star.  This  method  was  used  in  the  field  for  a  number  of  years. 


Form  256.* 


Computation  of  time  set. 

[Station,  Key  West,  Florida.    Date,  Feb.  14, 1907.    Set,2.    Observer,  3.  S.  Hill.    Computer,  J.  S.  Hill.] 


Star 


1.  S  Monocer. 

2.  <j>5  Aurigae 

3.  18  Monocer. 

4.  6  Geminor. 

5.  £  Geminor. 

6.  63  Aurigae 

7.  t  Geminor. 

8.  j9  Can.  Min. 

9.  a  Can.  Min. 

10.  /?  Geminor. 

11.  ic  Geminor. 

12.  <j>  Geminor. 


Clamp 


W 

w 

W 

w 
w 
w 

E 
E 
E 
E 
E 
E 


s 

+15.00 
+15. 08 
+15. 04 
+  15.03 
+15.00 
+15. 02 

+  14.43 
+14. 45 
+  14.45 
+14.41 
+14.42 
+14.47 


0.00 
+0.08 
+0.04 
+0.03 

0.00 
+0.02 

-0.57 
-0.55 
-0.55 
-0.59 
-0.58 
-0.53 


+  1.02 
+1.38 
+  1.01 
+  1.21 
+1.07 
+1.30 

-1.13 
-1.02 
-1.01 
-1.13 
-1.21 
-1.12 


+0.26 
-0.45 
+0.37 
-0.20 
+0.07 
-0.34 

-0.07 
+0.28 
+0.33 
-0.08 
-0.19 
-0.05 


Cc 


s 

+0.27 
+0.36 
+0.26 
+0.32 
+0.28 
+0.34 

-0.30 
-0.27 
-0.26 
-0.30 
-0.32 
-0.29 


Aa 


+0.02 
-0.03 
+0.03 
-0.01 
0.00 
-0.02 

0.00 
+0.01 
+0.01 

0.00 
-0.01 

0.00 


(a-0- 
Cc-Aa 


Mean  AT= 


+14.  71 
+  14.75 
+  14.75 
+  14.72 
+  14.72 
+14. 70 

+14. 73 
+14.71 
+14.  70 
+14.71 
+  14.75 
+  14.76 

.727 


1.  3.00   (M+3. 10  c+0. 70  ow  -0.04=0 

2.  3.00   <?t+3.89  c-0.99  ow  -0.13=0 

5.  2.12   3t+2. 75  c-0. 70  aw  -0.09=0 

6.  5.12   54+5.85  c  -0.13=0 
9.  4.71   cM+5.38  c  -0.12=0 

10.  9.53    Si  +2.61=0 


(2)X.707 
(6)X-920 


11.       9t= -0.274 


^r=+15.00-0.274=+14.726 


3.  3.00   5(-3. 15  c+0.  56  a  +1.63=0 

E 

4.  3.00   «-3.47   c-0. 34  a£  +1.74=0 

7.  1.82   di-1.91  c+0. 34  a£  +0.99=0 

8.  4.82   <5<-5.38  c  +2.73=0 
12.  -1.32  -5.38  c  +2.73=0 

14.  -0.82  +1.02     -0.99  «w  -0.13=0 

16.  -0.82  -0.83     +0.56  a.  +1.63=0 


(3)X-607 


13.        c= +0.262 


15.     aw= +0.071 


17. 


= +0.036 


+  .02 
-.02 
-.02 
+.01 
+.01 
+.03 

.00 
+.02 
+.03 
+  .02 
-.02 
-.03 


*  See  note  below  table  on  p.  18. 


DETERMINATION   OF   TIME.  27 

EXPLANATION   OF  ABOVE   COMPUTATION. 

The  serial  numbers  indicate  the  order  of  the  various  steps  of  the  computation. 
Each  equation,  for  a  star,  is  of  the  form: 


Equation  1  is  obtained  by  adding  corresponding  terms  of  the  three  such  observation  equa- 
tions for  the  three  south  stars  (1,  3,  and  5).  Equations  2,  3,  and  4  are  obtained  in  a  similar 
manner,  there  being  two  equations  in  each  half  set,  one  involving  the  three  stars  farthest  south, 
the  other  the  remaining  stars  of  the  half  set,  in  this  case  three  in  number.  There  are  then  four 
equations,  involving  four  unknowns,  which  can  be  solved  by  simple  algebraic  elimination.  In 
the  above  computation  this  has  been  reduced  to  systematic  mechanical  operations.  The 
azimuth  constants  are  first  eliminated,  next  c  is  eliminated,  and  then  dt  is  obtained.  The 
computation  is  so  arranged  that  the  multipliers  are  always  less  than  unity,  which  are  used 
to  reduce  coefficients  in  certain  equations  to  equality  with  corresponding  coefficients  in  other 
equations.  This  makes  it  possible  to  carry  through  the  entire  computation  with  the  aid  of 
Crelle's  (or  other  similar)  tables.  In  making  substitutions  in  equations,  such  as  14  and  16, 
where  there  is  a  choice  between  two  equations,  it  is  always  well  to  select  the  equation 
having  the  larger  coefficient  for  the  unknown  sought.  If  the  computation  is  followed  in 
these  respects  and  a  sufficient  number  of  whole  seconds  are  dropped  from  the  (oc  —  f)  to  insure 
that  dt  will  be  less  than  one  second,  there  is  no  necessity,  in  any  given  case,  of  carrying  the 
computation  to  a  greater  number  of  decimal  places  than  are  shown  above. 

The  checks  which  must  be  satisfied,  if  the  computation  is  correct,  are:  (1)  The  algebraic 
sum  of  all  the  residuals  must  not  in  hundred  ths  of  seconds  be  more  than  one-half  the  number 
of  stars  in  the  complete  set;  (2)  the  sum  of  the  two,  three,  or  four  residuals  corresponding  to 
each  of  the  four  equations  designated  above  as  1,  2,  3,  and  4  must  seldom  be  as  large  as,  and 
never  exceed,  Os.02. 

If  these  checks  are  not  satisfied,  the  following  principle  may  be  found  useful  in  detecting 
whether  the  error  was  made  during  the  process  of  solution  of  the  four  equations.  If  the  work 
of  solution  is  correct,  the  derived  values  of  the  unknowns  substituted  in  any  one  of  the  equations 
should  give  a  residual  not  greater  than  CP.Ol  (the  substitution  being  carried  to  thousandths  of 
seconds),  but  if  any  equation  shows  a  residual  greater  than  this,  the  error  in  the  solution  was 
made  in  deriving  an  equation  of  a  higher  serial  number,  the  serial  numbers  having  been  assigned 
in  the  order  in  which  the  computation  was  made. 

The  chronometer  correction  JJ1  is  then  equal  to  dt  plus  the  number  of  whole  seconds 
which  were  dropped  from  (ce  —  t)  in  order  to  lighten  the  work  involved  in  making  the  computa- 
tion. In  this  case  it  is  equal  to  —  08.274  +  158.00=  +148.726.  The  chronometer  epoch  for 
which  this  correction  applies  is  the  mean  of  the  chronometer  times  of  the  observed  transits;  that 
is,  the  mean  of  the  t's.  It  is  not  the  mean  of  the  right  ascensions  —  unless,  of  course,  the  chronom- 
eter correction  happens  to  be  zero. 

While  it  is  advisable  to  have  the  instrumental  constants  c,  a^,  and  0%  small,  it  is  not 
desirable  to  strive  to  have  them  close  to  zero.  For  the  azimuth  constant  one  second  is  a  good 
limit  to  keep  within,  while  if  the  collimation  constant  is  less  than  0s.  2  it  is  well  not  to  attempt 
further  adjustment  with  a  view  of  reducing  it. 

The  computations  are  somewhat  simpler  when  the  transit  is  reversed  on  each  star  and  one- 
half  the  observations  on  a  star  are  made  in  each  of  the  positions  —  band  west  and  band  east  — 
for  the  collimation  is  eliminated  by  the  method  of  observing  and  the  only  unknowns  are  one 
azimuth  constant  and  the  clock  correction,  AT. 


28  U.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION   NO.   14. 

A   SECOND   EXAMPLE   OF  RECORD  AND   COMPUTATION. 

On  page  26  reference  is  made  to  a  second  method  of  solution  for  AT,  a,  and  c,  without 
the  use  of  least  squares.  This  second  method  is  used  when  a  different  selection  of  stars  is  made 
from  that  shown  on  page  18.  The  difference  between  the  two  star  sets  is  that  in  the  example 
of  computation  shown  on  page  26  the  instrumental  constants  c  and  a  are  determined  from  all 
the  stars,  each  star  being  given  unit  weight,  while  in  the  method  which  follows  there  is  observed 
in  each  half  set  a  slow-moving  star,  called  the  azimuth  star,  from  which  the  azimuth  constant 
for  that  half  set  is  principally  determined.  Besides  this  azimuth  star  there  are  four  time  stars 
in  each  half  set,  and  it  is  from  the  eight  time  stars  in  the  entire  set  that  the  collimation  constant 
is  mainly  derived.  It  seems  that  the  method  of  having  all  time  stars  in  a  set  is  preferable  to 
the  other  method,  in  which  both  time  and  azimuth  stars  are  used.  In  the  former,  the  clock 
correction  depends  on  all  12  stars  instead  of  being  derived  mainly  from  8  stars  only,  and 
the  collimation  correction  is  more  accurately  determined.  The  azimuth  constants,  however, 
are  not  so  accurately  determined  by  the  first  as  by  the  second  method,  but  this  is  immaterial 
if  the  plus  and  minus  azimuth  factors  in  each  half  set  are  about  equally  balanced. 

While  this  second  method  has  been  superseded  in  the  longitude  work  of  the  Coast  and 
Geodetic  Survey,  it  is  considered  desirable  to  continue  it  in  this  publication. 

Using  this  second  method,  time  acceptable  for  latitude  or  azimuth  work  can  be  easily 
obtained  with  a  meridian  telescope,  a  zenith  telescope,  or  even  with  an  engineer's  transit  or 
theodolite.  In  its  usual  form  the  star  set  consists  of  four  tune  stars  and  an  azimuth  star  with 
the  instrument  in  each  position,  band  west  and  band  east.  If  greater  accuracy  is  desired  the 
number  of  time  stars  in  a  half  set  may  be  increased,  or  if  less  accuracy  is  needed  the  number  may 
be  decreased.  In  the  work  of  the  Survey  up  to  the  time  of  the  adoption  of  the  transit  micrometer 
and  the  method  of  computation  shown  on  pages  20-27,  the  standard  time  set  consisted  of  two 
half  sets,  in  each  of  which  was  one  azimuth  star  and  four  time  stars. 

The  following  set  of  observations  was  made  with  a  small  portable  transit,  using  an  observing 
key  to  record  the  observations  chronographically.  With  the  record  of  observations  there  are 
given  the  readings  of  the  level,  the  correction  for  inclination  of  the  horizontal  axis  of  the  tele- 
scope (which  in  this  case  includes  a  correction  for  inequality  of  pivots),  and  the  computation  of 
(«-<).  A  correction  for  rate  has  been  introduced.  The  correction  for  diurnal  aberration  and 
the  correction  for  rate  are  obtained  in  the  same  manner  as  shown  on  page  24.  The  form  on 
which  the  level  readings  are  recorded  is  shown  on  page  20. 


DETERMINATION   OF   TIME. 
Star  list  for  Washington,  D.  C. — Latitude  38  °  54'  N. 


29 


Star  factors 

Star 

Cata- 
logue 

Magni- 
tude 

Right  ascen- 
sion 
a 

Declina- 
tion 

i 

Zenith  dis- 
tance 

C 

Diurnal 
aberra- 
tion 

Incli- 
na- 

Colli- 
ma- 

Azimuth 

K 

tion 

tion 

A 

B 

C 

h    m      s 

o        / 

o        / 

s 

17  H.  Can.  Yen. 

B 

5.5 

13     30     12 

+37     43 

+   1      11 

-.02 

1.26 

1.26 

+  .02 

t)  Ursa  Maj. 

B 

2.0 

43     30 

+49    50 

-10    56 

-.02 

1.53 

1.55 

-  .30 

rj  Bootis 

B 

3.0 

49     47 

+18    55 

+19    59 

-.02 

0.99 

1.06 

+  .36 

11  Bootis 

B 

6.0 

56    31 

+27     53 

+11    01 

-.02 

1.11 

1.13 

+  .22 

a  Draconis 

B 

3.3 

14    01    39 

+64     52 

-25    58 

-.04 

2.12 

2.36 

-1.03 

d  Bootis 

B 

5.0 

05    42 

+25    35 

+13     19 

-.02 

1.08 

1.11 

+  .25 

or  Bootis 

B 

1.0 

10    58 

+19    43 

+19    11 

-.02 

1.01 

1.06 

+  .35 

A  Bootis 

B 

4.0 

12    29 

+46    34 

-  7    40 

-.02 

1.44 

1.46 

-  .19 

d  Bootis 

B 

3.8 

21    43 

+52    20 

-13    26 

-.03 

1.59 

1.64 

-  .38 

5  Ursse  Min. 

A 

4.5 

27    51 

+76    09 

-37    15 

-.06 

3.33 

4.18 

-2.53 

B-=  Berliner  Astronomisches  Jahrbuch.    A=American  Ephemeris. 


30 


U.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.   14. 


Following  the  computation  are  given  any  explanations  needed  to  supplement  or  qualify 
the  explanations  of  computations  given  on  pages  22-27. 


[Station,  Washington,  D.  C.    Date,  May  17, 1896.    Observer,  G.  R.  P. 


Star 

17  H.  Can.  Yen. 

,  Urs.  Maj. 

jj  Bootis 

n  Bootis 

a  Draco. 

Position  of  band 

Mrest 

West 

West 

West 

West 

Direction  of  ob- 

| 

jective  for  level 

* 

S 

JV 

reading 

J 

W                     E 

W                E 

W                E 

d                    d 

d                d 

d                d 

22.7                   24.1 

27.  8             20.  0 

28.  0               19.  9 

Level  readings 

27.  1                   19.  9 

22.  9             24.  8 

22.  9               24.  9 

SVt  and  JE 

49.8                  44.0 

50.  7             44.  8 

50.  9               44.  8 

JW-JE 

+5.8 

+5.9 

+6.1 

Remarks      and 

Means  of  levels 

comput  a  t  i  o  n 

d 

of  b 

N  +6.  10 

1 

S  +5.85 

+5.  98X.0279= 

+  .  167=0W 

—  .  010—  pivot  ine- 

quality 

+  .  157=bw 

Observed  transit 

h  m       t 

h  m       s 

h  m      s 

h  m       s 

h  m       s 

Line  1 

13  29  56.  90 

13  43  10.  60 

13  49  34.45 

13  56  17.20 

14  01  07.  55 

2 

13  30  00.  10 

Mean 

14.35 

37.05 

20.00 

13.30 

3 

03.30 

5 

18.30 

39.70 

22.80 

19.35 

4 

09.70 

14.20 

26.15 

45.00 

28.40 

31.25 

5 

12.90 

Correc- 

30.15 

j 

47.75         l 

31.50 

$ 

37.30 

s 

6 

16.00 

tion 

33.80 

33.80 

50.25 

50.25 

34.30 

34.30 

43.00 

43.00 

7 

19.30 

15.15 

37.95 

68.10 

52.95 

100.70 

36.90 

68.40 

49.00 

86.30 

8 

22.60 

41.70 

7.85 

55.70 

0.70 

40.05 

8.45 

55.20 

6.45 

9 

29.00 

10 

49.70 

8.00 

13  50  00.  90 

0.60 

45.65 

X.45 

14  02  06.  90 

6.25 

10 

32.20 

X  1.26 

53.60 

7.95 

03.75 

0.80 

48.60 

8.60 

12.90 

6.20 

11 

- 

-  +  1.92 

57.55 

8.15 

06.50 

0.95 

51.50 

8.70 

18.85 

6.40 

Mean 

16.12 

33.99 

10.85 

50.36 

4.00 

34.26 

2.90 

43.15 

1.60 

R 

+  .03 

+  .02 

+  .01 

+  .01 

.00 

K 

-  .02 

-  .02 

-  .02 

-  .02 

-  .04 

Bb 

+  .20 

+  .24 

+  .16 

+  .17 

+  .33 

t 

13  30  16.  33 

13  43  34.  23 

13  49  50.51 

13  56  34.  42 

14  01  43.  44 

a 

13  30  12.  26 

13  43  30.  14 

13  49  46.  62 

13  56  30.53 

14  01  38.92 

OL-t 

-4.07 

-4.09 

-3.69 

-3.89 

-4.52 

DETERMINATION   OF   TIME. 


31 


Instrument,  transit  No.  18.    Chronometer,  Negus,  1836  (daily  rate,  1«.51  gaining).] 


d  Bootis                          a  Bootis 

A  Bootis 

e  Bootis 

5  Urs.  Min. 

East 

East 

East 

East 

East 

S 

N 

A 

W                     E 

W                          E 

W                       E 

d                      d 

d                           d 

d                      d 

27.1                  20.9 

27.  2                        20.  9 

22.  2                     26.  0 

22.7                   25.2 

22.9                        25.3 

27.2                   21.0 

49.  8                    46.  1 

50.  1                         46.  2 

49.  4                     47.  0 

+3.7 

+3.9 

+2.4 

Means  of  levels 

Thin  clouds  and  hazy 

d 

Temperature  76°  F 

N.  +3.  15 

S.  +3.70 

j 

1 

+3.42X.0279- 

+  .095-#E 

+  .  010=  pivot  inequality 

+.105=bE 

Am       s 
14  05  29.  40 

Mean 

Am      j 
14  10  45.50 

Am      s 
14  12  11.  15 

Am      s 
14  21  22.30 

Am       s 
14  26  53.  15 

32.20 

s 

48.20 

14.80 

26.40 

27  03.15 

34.85 

44.76 

50.90 

18.60 

30.60 

14.25 

40.60 

Correc- 

56.20 

25.95 

38.90 

35.30 

43.35 

tion 

58.90 

1 

29.50 

J 

42.90 

1 

45.85 

s 

46.20 
48  90 

12.69 

14  11  01.65 
04  30 

01.65 
03  20 

33.45 

37  00 

33.45 
66  50 

47.35 
51  35 

47.35 

94  25 

57.15 

UOQ    1-17    fu-1 

S7.15 

119  Sfi 

51.90 

10 

07.10 

3.30 

40.70 

6.65 

55.40 

4.30 

£>O   U<  .  UU 

18.00 

I  1_.  BO 

3.30 

57.30 

X  1.11 

12.30 

3.20 

48.00 

6.60 

14  22  03.  60 

4.20 

38.70 

2.95 

- 

-   1.41 

15.20 

3.40 

51.80 

6.60 

07.80 

4.20 

49.50 

2.65 

14  06  02.  90 

17.70 

3.20 

55.25 

6.40 

11.95 

4.25 

14  29  00.  05 

3.20 

46.17 

01.63 

6.95 

33.29 

3.20 

47.14 

1.55 

56.55 

72.10 

.00 

-  .01 

-  .01 

-  .02 

-  .03 

-  .02 

-  .02 

-  .02 

-  .03 

-  .06 

+  .11 

+  .11 

+  .15 

+  .17 

+  .35 

14  05  46.26 

14  11  01.71 

14  12  33.  41 

14  21  47.26 

14  27  56.  81 

14  05  42.32 

14  10  57.  90 

14  12  29.  18 

14  21  42.  97 

14  27  51.37 

-3.94 

-3.81 

-4.23 

-4.29 

-5.44 

32 


U.   S.   COAST   AND   GEODETIC   SUBVEY   SPECIAL   PUBLICATION    NO.    14. 


REDUCTION  OF  INCOMPLETE  TRANSITS. 

If  the  transit  of  a  star  across  every  line  of  the  diaphragm  is  observed,  the  mean  of  the 
times  is  the  required  time  of  transit  across  the  mean  line.  In  obtaining  the  sum  of  the  several 
observed  times  any  gross  error  in  any  one  of  the  times  may  be  detected  by  using  the  auxiliary 
sums,  shown  in  the  example  on  pages  30-31,  in  the  little  column  just  after  the  observed  times, 
namely,  the  sum  of  the  first  and  last  times,  of  the  second  and  last  but  one,  third  and  last  but 
two,  etc.  These  auxiliary  sums  should  be  nearly  the  same  and  nearly  equal  to  double  the  time 
on  the  middle  line.  This  is  also  a  convenient  method  of  taking  means,  as  it  is  in  general  only 
necessary  to  sum  the  decimal  columns. 

When  the  star  was  observed  on  some  of  the  lines  but  missed  upon  the  others,  the  time  of 
transit  over  the  mean  of  all  the  lines  may  be  found  as  follows: 


tm  =  mean  of  observed  times  — 


(sum  of  equatorial  intervals  of  observed  lines)   (sec 
number  of  observed  lines. 


or 


(sum  of  equatorial  intervals  of  missed   lines)    (sec   S) 
=  mean  of  observed  times  +  -  number  ofobserved  line^T 


The  first  of  these  formulae  is  the  more  convenient  if  but  few  lines  were  observed  and  the 
second  the  more  convenient  if  but  few  lines  were  missed.  The  two  incomplete  transits  shown 
in  the  example  on  pages  30-31  were  reduced  by  the  second  formula. 

tm  is  the  time  of  transit  across  the  mean  of  all  the  lines  of  the  diaphragm.  The  equatorial 
interval  of  a  given  line  is  the  time  which  would  elapse  between  the  transit  of  an  equatorial  star 
over  the  mean  line  of  the  diaphragm  and  the  transit  over  the  line  in  question.  It  is,  in  seconds 
of  time,  ^  the  angular  interval  between  the  lines  expressed  in  seconds  of  arc.  An  equatorial 
interval  is  called  positive  when  the  transit  across  the  line  in  question  occurs  later  than  the  transit 
across  the  mean  line.  The  signs  of  all  the  equatorial  intervals  are  therefore  reversed  when  the 
horizontal  axis  of  the  telescope  is  reversed. 

For  an  example  of  the  method  of  computing  the  equatorial  intervals  see  page  44. 

The  above  formulae  for  reduction  to  the  mean  line  are  approximate,  and  the  maximum 
possible  error  of  the  approximation  increases  with  an  increase  in  the  declination  of  the  star 
and  with  an  increase  in  the  equatorial  intervals  of  the  extreme  lines.  If  the  extreme  equatorial 
interval  is  60s,  the  maximum  error  is  less  than  08.01  for  a  star  of  which  <?  =  70°,  and  is  only 
03.3  if  5  =  85°.  If  the  extreme  interval  is  15s,  the  maximum  error  is  less  than  08.01  if  «J  =  85°. 

The  more  exact  formula  for  use  with  circumpolar  stars  is  the  same  as  that  given  above, 
except  that  for  each  equatorial  interval,  i,  must  be  substituted  i  %j  sec  r,  in  which  r  is  the  hour 
angle  of  the  star  at  transit  across  the  line,  or  with  sufficient  accuracy  r  =  i  sec  3  =  the  actual  time 
interval  from  the  mean  line. 

The  following  table  will  be  found  useful  in  connection  with  this  formula. 


T 

log  Veos  t 

log  V  sec  T 

T 

T 

log  V  sec  T 

log  V  COS  r 

log  V  sec  i 

log  V  COS  T 

m 

TO 

TO 

1 

9.  99999 

0.00000 

16 

9.  99965 

0.00035 

31 

9.  99867 

0.  00133 

2 

99 

01 

17 

960 

040 

32 

858 

142 

3 

99 

01 

18 

955 

045 

33 

849 

151 

4 

98 

02 

19 

950 

050 

34 

840 

160 

5 

97 

03 

20 

945 

055 

35 

831 

169 

6 

95 

05 

21 

939 

061 

36 

821 

179 

7 

93 

07 

22 

933 

067 

37 

811 

189 

8 

91 

09 

23 

927 

073 

38 

800 

200 

9 

89 

11 

24 

921 

079 

39 

789 

211 

10 

86 

14 

25 

914 

086 

40 

778 

222 

11 

83 

17 

26 

907 

093 

41 

767 

233 

12 

80 

20 

27 

899 

101 

42 

756 

244 

13 

77 

23 

28 

892 

108 

43 

744 

256 

14 

73 

27 

29 

884 

116 

44 

732 

268 

15 

9.  99969 

0.00031 

30 

9.  99876 

0.  00124 

45 

9.  99719 

0.  00281 

'  The  collimation  factor  C  (as  given  in  the  star  list  on  p.  29)  is  the  sec  i. 


DETERMINATION   OF   TIME. 


33 


If  the  chronometer  rate  exceeds  15s  per  day  it  will  be  desirable  to  take  it  into  account  in 
making  the  reduction  of  incomplete  transits  to  the  mean  line. 

Another  method  of  reducing  incomplete  transits  is  to  construct  from  the  known  equatorial 
intervals  a  table  similar  to  that  of  which  a  portion  is  printed  below  showing  the  interval  of  each 
line  from  the  mean  line  corresponding  to  various  declinations.  The  correction  of  each  observed 
line  to  the  mean  line  is  then  taken  out  directly  from  the  table  and  the  mean  of  the  various 
corrected  transits  taken. 

Intervals  of  lines  of  Transit  No.  18  from  mean  line. 

[The  numbering  of  the  lines  is  for  band  west.] 


3 

Line  I 

Line  II 

Line  III 

Line  IV 

LineV 

Line  VI 

Line  VII 

Line  VIII 

Line  IX 

LineX 

Line  XI 

o 

0 
10 
15 

s 
+15.  20 
15.43 
15.74 

s 
+12.  69 
12.89 
13.14 

S 
+10.  15 
10.31 
10.51 

s 
+5.06 
5.14 
5.24 

S 
+2.52 
2.56 
2.61 

-0.09 
0.09 
0.09 

S 
-2.52 
2.56 
2.61 

* 
-5.11 
5.19 
5.29 

t 
-10.09 
10.25 
10.45 

-12.65 
12.84 
13.10 

s 
-15.15 
15.38 
15.68 

36 
38 

40 

18.79 
19.29 
19.84 

15.69 
16.10 
16.57 

12.55 
12.88 
13.25 

6.25 
6.42 
6.61 

3.11 
3.20 
3.29 

0.11 
0.11 
0.12 

3.11 
3.20 
3.29 

6.32 
6.48 
6.67 

12.47 
12.80 
13.17 

15.64 
16.05 
16.51 

18.73 
19.23 
19.78 

51 
52 
53 

24.15 
24.69 
25.26 

20.17 
20.61 
21.09 

16.13 
16.49 
16.87 

8.04 
8.22 
8.41 

4.00 
4.09 
4.19 

0.14 
0.15 
0.15 

4.00 
4.09 
4.19 

8.12 
8.30 
8.49 

16.03 
16.39 
16.77 

20.10 
20.55 
21.02 

24.07 
24.61 
25.17 

Transit  No.  18  was  the  instrument  used  for  the  observations  shown  on  pages  30-31.  The 
incomplete  transit  of  the  star  17  H.  Can.  Ven.,  of  which  the  declination  is  37°  43',  may  be 
computed  as  indicated  below: 


Line 
I 

II 

III 

IV 

V 

VI 

VII 

VIII 

IX 

X 


Correction 
+19.  22 
+16.04 
+12.  83 
+  6.40 
+  3.19 

-  0.11 

-  3.19 

-  6.46 
-12.75 
-15.99 


Corrected  transit 
16.12 
16.14 
16. 13 
16.10 
16.09 
15.89 
16.11 
16.14 
16.25 
16.21 


Mean  =16. 12,  agreeing  with  the  result  shown  in  the 
example  on  page  30. 


The  special  advantage  of  this  method  of  reducing  incomplete  transits  is  that  a  wild  observa- 
tion upon  any  one  line  is  at  once  detected.  Such  wild  observations  are  apt  to  occur  under  the 
conditions  which  produce  incomplete  transits,  viz.,  clouds,  haste,  or  difficulty  with  illumination. 

CORRECTION  FOR  RATE. 
The  method  of  computing  this  correction  is  shown  on  page  24. 

CORRECTIONS  FOR  DIURNAL  ABERRATION,  COLLIMATION,  AND  AZIMUTH. 

The  correction  for  diurnal  aberration  and  general  expressions,  for  the  collimation  and 
azimuth  corrections  are  shown  on  pages  24-25. 
8136°— 13 3 


34  U.    S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.   14. 

COMPUTATION  OF  4  T,  a  AND  c,  USING  AZIMUTH  STARS  AND  METHOD  OF  APPROXIMATIONS. 

The  method  of  computation  shown  below  was  in  use  in  the  field  by  parties  of  this  Survey 
for  many  years.1     It  is  now  replaced  by  the  method  shown  on  page  26. 

[Station,  Washington,  D.  C.    Date,  May  17, 1890.] 


Star 

Band  !    o.-t 

C 

A 

Cc 

Aa 

tt-t-  Cc-Aa 

s 

s 

s 

s 

s 

17  H.  Can.  Ven. 

W 

-4.07 

+  1.26 

+  .02 

+.04 

+  .01 

-4.12 

+  .10 

n  Urs.  Maj. 

W 

-4.09 

+1.55 

-  .30 

+.05 

-  .17 

-3.97 

-.05 

i)  Bootis 

W 

-3.69 

+1.06 

+  .36 

+.03 

+  .20 

-3.92 

-.10 

II  Bootis 

W 

—3.89 

+1.13 

+  .22 

+.04 

+  .12 

-4.05 

+  .03 

at  14^  02m.OO 

a  Draconis 

W 

-4.52 

+2.36 

-1.03 

+.08 

-  .58 

-4.02 

.00 

J  T=  -04«.024 

d  Bootis 

E 

-3.94 

-1.11 

+  .25 

-.04 

+  .13 

-4.03 

.00 

a  Bootis 

E 

-3.81 

-1.06 

+  .35 

-.03 

+  .18 

-3.S6 

-.07 

1  Bootis 

E 

-4.23 

-1.46 

-  .19 

-.05 

-  .10 

-4.08 

+.05 

0  Bootis 

E 

-4.29 

-1.64 

-  .38 

-.05 

-  .19 

-4.05 

+  .02 

S  Urs.Min. 

E 

-5.44 

-4.18 

-2.53 

-.13 

-1.28 

-4.03 

.00 

Or-< 

C 

A 

Cc 

a-t-  Cc 

Aa 

O.-t-  Cc-Aa 

First  approximation: 

s 

Mean  of  time  stars 

W 

-3.94 

+1.25 

+  .08 

+.06 

-4.00 

+  .05 

-4.05 

c=+.0ol 

Azimuth  star 

W 

-4.52 

+2.36 

-1.03 

+.12 

-4.64 

—  .59 

-4.05 

ow—  +  .577 

Mean  of  time  stars 

E 

-4.07 

-1.32 

+  .01 

-.07 

-4.00 

.00 

-4.00 

BE-+.484 

Azimuth  star 

E 

-5.44 

-4.18 

-2.53 

-.21 

-5.23 

-1.23 

-4.00 

Second  approximation 

Mean  of  time  stars 

W 

+  .04 

-3.98 

+  .04 

-4.02 

C=+.032 

Azimuth  star 

W 

+  .08 

-4.60 

-  .58 

-4.02 

ow=  +  .559 

Mean  of  time  stars 

E 

-.04 

-4.03 

.00 

-4.03 

nE_+.504 

Azimuth  star 

E 

-.13 

-5.31 

-1.28 

-4.03 

i 

1  Thecomplete  formula  for  the  chronometer  correction  is  A  T=<x.-(tm+R+K+Bb+Cc+Aa).    Let  t=tm+R+K+£b,  then  A  r=(a— 0—  Cc-Aa 
so  that  it  will  be  seen  that  the  corrections  Cc  and  Aa  are  to  be  subtracted  algebraically  from  a— (. 

EXPLANATION  OF  THE  COMPUTATION. 

The  first  five  columns  of  the  upper  portion  of  the  computation  were  compiled  from  the 
record  and  computation  shown  on  pages  30-31  and  from  the  observing  list  shown  on  page  29, 
The  remaining  columns  were  filled  out  after  the  computation  of  a  and  c,  shown  in  the  lower 
portion  of  the  form,  was  completed. 

It  should  be  noted  that  the  five  stars  of  each  group,  observed  in  one  position  of  the  instru- 
ment, have  been  so  selected  that  one  is  a  slowly  moving  northern  star  at  a  considerable  distance 
from  the  zenith,  while  the  other  four  are  all  comparatively  near  the  zenith,  some  transiting  to 
the  northward  of  it  and  some  to  the  southward,  and  at  such  distances  from  it  that  then-  mean 
azimuth  factor,  A,  is  nearly  zero.  These  four  stars  of  each  group  may  be  for  convenience  called 
time  stars,  since  the  determination  of  time  falls  mainly  upon  them,  while  the  slowly  moving 
northern  star  serves  to  determine  the  azimuth  error  of  the  instrument,  and  may  be  called  the 
azimuth  star. 

In  this  computation  to  derive  c  and  a  the  time  stars  in  each  position  of  the  instrument 
are  combined  and  treated  as  one  star  by  taking  the  means  of  their  (n  —  t)'&,  and  of  their  star 
factors  C  and  A,  respectively,  these  means  being  written  below  the  separate  stars  in  the  form, 
together  with  the  azimuth  stars.  On  the  assumption  that  the  means  of  the  time  stars  in  the 
two  positions  of  the  instrument  are  equally  affected  by  the  azimuth  correction,  the  first  approxi- 

*  It  was  devised  in  the  seventies  by  Assistant  Edwin  Smith,  then  an  aid  in  this  Survey.    See  p.  280,  Appendix  4  of  the  Report  for  1904. 


DETERMINATION    OF   TIME.  35 

mation  to  c  is  found  by  dividing  the  difference  between  tbe  mean  (a  —  t)'s  by  the  difference 
between  the  6"s.  In  the  example, 

,.      .      (.-*-t)w-((Y-t)E     -3.94-  (-4.07)      +0.13 
c  (hist  approximation)  =        -Q^C^       =  +  1.25-7^.32)  =  +2^7  =  +  °  -051' 

Tsing  this  approximation  to  c,  the  correction  Cc  is  then  subtracted  from  the  a  —  t  of  each  mean 
of  the  time  stars  and  of  each  azimuth  star,  and  the  values  of  a  —  t—  Cc,  in  the  seventh  column 
on  the  fifth  to  eighth  lines  from  the  bottom  of  the  form,  are  obtained. 

Separate  values  for  the  azimuth  error  of  the  instrument  are  then  derived  for  each  position 
of  the  instrument  as  follows: 

(  T  -  t  -  (7c)tlme  stars  ~(ne-t-  fle)a,|muth  star       -  4.00  -(-  4.64)        +0.64 
' 


time  stars 


= 
-A  azimuth  star.  "  +0.08  -  (-  1.03)  =  +  1.11 


-4.00  -(-5.23)      +1.23 
«*=  VOTOI—  C-2T53r  +2754=      * 

With  these  values  of  aw  and  <IE  the  corrections  Aa  are  applied,  giving  the  values  ct  —  i—  Cc  —  Aa 
in  the  last  column  but  one.  If  these  do  not  agree  for  the  stars  east  and  west  it  indicates  that 
the  mean  values  ce  —  t,  used  in  deriving  c,  were  not  equally  affected  by  the  azimuth  error,  so  that 
their  difference  was  not  entirely  due  to  c,  as  was  assumed.  An  improved  value  of  c  may  now 
be  obtained  by  treating  the  difference  in  the  last  column  as  still  an  error  of  collimationj  and 
thus  obtaining  a  correction  to  the  first  approximate  value  of  c.  Thus,  in  the  example, 

-4.05  -(-  4.00)  _  -0.05  _ 
+  1.25-  (-1.32)      +2.57 

Applying  this  correction  to  the  first  approximate  value  of  c=  +0.051,  we  have  for  a  second 
approximation  c=  +0.032.  Proceeding  as  before,  improved  values  for  aw  and  aE  are  found. 
If  the  star  sets  are  well  chosen  and  the  instrumental  errors  small,  the  first  approximation  will 
generally  suffice.  If  the  values  of  a  —  t—  Cc  —  Aa  differ  by  but  a  few  hundredths,  east  and  west, 
there  is  little  gained  by  making  a  closer  adjustment.  The  chronometer  correction  will  prob- 
ably not  be  changed  at  all,  but  the  instrumental  errors  and  star  residuals  will  be  slightly  altered, 
as  is  apparent  from  the  example,  where  the  closer  adjustment  is  made  for  the  purpose  of  illus- 
trating the  method. 

In  the  first  approximation  the  value  of  c  may  at  once  be  derived  more  closely  when  there 
is  much  difference  between  the  mean  A's  for  the  time  stars,  by  estimating  the  effect  of 
this  difference  in  A  on  the  A  T,  and  allowing  for  this  effect  when  deriving  c  in  the  first  place. 
The  formula  for  c  then  becomes 


_ 
c~ 


It  is  here  necessary  to  estimate  the  azimuth  of  the  instrument,  a,  roughly  in  advance,  and 
this  may  be  done  by  inspection.  Thus,  in  the  example,  assuming  a=  +08.5,  we  have 

_  -3.94-^4.07-  (+.  07)  X  (  +  0.5)  _  +.09  _ 
+  1.25  +  1.32  =+2.57~ 

agreeing  closely  with  the  value  j^iven  by  the  second  approximation. 

When  satisfactory  values  of  c,  aw,  and  «E  have  been  obtained,  the  corrections  Cc  and  Aa 
are  applied  separately  to  each  star,  as  shown  in  the  upper  part,  and  the  values  of  the  chronometer 
correction  (AT)  derived  separately.  The  residuals  are  taken  for  each  group  from  the  mean 
of  that  group,  and  thus  furnish  a  convenient  check  on  the  computation,  as  their  sums  for  each 
group  should  approximate  zero.  Unusual  residuals  also  point  to  possible  errors  in  a  —  t.  The 


36 


TJ.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.   14. 


mean  of  the  A  T's  from  the  separate  stars  gives  the  final  chronometer  correction  at  the  epoch  of 
the  mean  of  the  chronometer  times  of  transit  of  the  stars  observed. 

This  whole  computation  may  be  made  with  rapidity  by  the  use  of  Crelle's  multiplication 
tables. 

The  field  computation  having  been  made  as  outlined  above,1  the  more  refined  office  com- 
putation may  be  made  as  indicated  on  pages  39-41.  It  is  desirable  in  this  office  computation 
to  introduce  weights  dependent  upon  the  declination  of  the  star  and  the  number  of  lines  of  the 
reticle  upon  which  the  star  was  observed. 

The  four  equations,  solved  by  successive  approximations  above,  may  be  solved  by  direct 
elimination,  in  case  the  coefficients  of  aw  and  aE  do  not  become  relatively  small  in  the  two  equa- 
tions gotten  by  taking  the  mean  of  the  time  stars  in  the  two  half  sets. 

RELATIVE    WEIGHTS    FOR    INCOMPLETE    TRANSITS. 

Sometimes  the  transit  of  a  star  is  observed  over  some  of  the  lines  of  the  diaphragm  and 
missed  over  the  others.  Obviously  the  deduced  time  of  transit  over  the  mean  line  from  such 
an  incomplete  transit  should  be  given  less  weight  than  that  from  a  complete  transit. 

For  observations  made  by  the  eye  and  ear  method  the  relative  weights  given  by  Chauvenet 
may  be  used,  viz: 

n  (N+3) 
P~  N  (n  +  3) 

in  which  p  is  the  weight  to  be  assigned  to  the  computed  time  of  transit  over  the  mean  line,  N 
is  the  total  number  of  lines  in  the  diaphragm,  and  n  is  the  number  of  lines  upon  which  obser- 
vations were  made.2  This  formula  is  based  upon  the  assumption  that  (c)2  =  3(£,)2,  in  which  (E)  = 
the  probable  error  of  an  observed  transit  of  an  equatorial  star  over  a  single  line  and  (e,)  =the 
probable  culmination  error  referred  to  the  equator,  a  constant  for  all  the  fines  of  the  diaphragm 
for  any  one  star,  but  variable  from  star  to  star,  and  supposed  to  be  due  mainly  to  atmospheric 
displacement,  to  outstanding  instrumental  errors,  to  irregularities  in  clock  rate,  and  to  changes 
in  personal  equation. 

The  following  table  shows  the  values  of  p  and  V?  for  the  two  cases  of  5  and  7  fines  in  the 
diaphragm : 

Table  of  weights  for  incomplete  transits  for  use  with  eye  and  ear  observations. 


N=S 

N=7 

P 

Vp 

P 

Vp 

1 

0.40 

0.63 

0.36 

0.60 

2 

0.64 

0.80 

0.57 

0.75 

3 

0.80 

0.89 

0.71 

0.84 

4 

0.92 

0.96 

0.82 

0.91 

5 

1.00 

1.00 

0.  90            0.  95 

6 

0:  95            0.  97 

7 

1.00 

1.00 

i  For  a  more  complete  account  of  this  method  of  computation,  see  Appendix  No.  9,  Report  for  1896.  The  above  account  is  largely  taken  from 
that  appendix. 

»  See  Chauvenet's  Astronomy,  Vol.  II,  p.  198.  The  derivation  of  this  formula  follows  the  same  lines  as  that  given  on  the  following  pages  for 
weights  to  be  assigned  to  incomplete  transits  taken  by  the  chronographic  method. 


DETERMINATION   OF   TIME.  37 

The  relative  weights  to  be  assigned  to  incomplete  transits  observed  by  the  chronograph 
method  may  be  derived  as  follows : 

r2=(E1)2  +  i^ 

in  which  r  =  the  probable  error  of  the  time  of  transit  over  the  mean  line,  arising  from  the  com- 
bined effect  of  the  culmination  error  referred  to  the  equator  (EJ)  and  of  the  probable  error  of 
the  transit  of  an  equatorial  star  over  a  single  line  (E). 

To  find  r,  individual  determinations  of  right  ascensions  of  stars,  all  referred  to  the  same 
epoch  (mean  place),  may  be  compared  with  their  respective  average  values;  thus,  from  558 
results  of  36  stars  observed  at  the  United  States  Naval  Observatory  with  the  transit  circle 
(using  a  magnifying  power  of  186)  in  1870  and  1871,  it  was  found  that  r=  ±08.034.  To  apply 
tliis  value  to  our  instruments  it  must  be  somewhat  increased,  though  not  in  proportion  to  the 
respective  magnifying  powers,  since  some  of  the  errors  involved  approach  the  character  of 
constants ;  multiplying  it  by  1 .5  and  1 .75  for  our  larger  and  smaller  transits,  respectively,  there 
is  obtained  r=  ±08.051  and  r=  ±08.060.  For  the  larger  transits  (E)=±08.063  and  for  the 
smaller  («)=  ±08.080.  (See  p.  39.)  Substituting  these  values  in  the  above  formula,  together 
with  the  values  25  and  15  for  n  as  actually  used  in  the  observations  cited  on  page  38,  there  is 
obtained 


(0.051)2=  (0*  +  and  (0-060)'=  (0'  + 

which  give 

(O  =  ±09.049  and  (E,)  =  ±0".  056 

for  the  larger  and  smaller  instruments,  respectively. 

If  the  weight  for  a  complete  transit  is  unity,  the  weight  for  an  incomplete  transit  is 


Hence,  for  the  larger  instruments,  using  the  above  values  for  (E,)  and  (E), 


and  for  the  smaller  instruments 

2.0 


n 


very  nearly.  From  these  expressions  the  relative  weights  have  been  computed  for  total  number 
of  threads  N=25,  17,  13,  and  11  for  the  larger  instruments  and  for  N=15,  13,  11,  and  9  for 
the  smaller  ones,  and  are  shown  in  the  following  table. 


38  U.   S.    COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.    14. 

Table  of  weights  for  incomplete  transits  for  use  with  chronograj>hic  observations. 


Number 
o'lines 
» 

For  large  portable  transits 

For  small  portable  transits 

JV-25 

JV=17 

N-  13 

ff~n 

2V-  15 

W=13 

jy-ii 

AT=9 

P 

VP 

P 

VP" 

P 

VP 

p 

VP~ 

P 

VP" 

P 

VP 

p 

VP~ 

P 

VP 

1 

.41 

.64 

.42 

.65 

.43 

.66 

.44 

.66 

.38 

.62 

.38 

.62 

.39 

.63 

.41 

.64 

2 

.59 

.77 

.61 

.78 

.62 

.79 

.64 

.80 

.56 

.75 

.58 

.76 

.59 

.77 

.61 

.78 

3 

.69 

.83 

.71 

.84 

.73 

.86 

.75 

.86 

.68 

.83 

.69 

.83 

.71 

.84 

.73 

.86 

4 

.76 

.87 

.78 

.88 

.80 

.90 

.82 

.90 

.75 

.87 

.77 

.88 

.79 

.89 

.82 

.90 

5 

.81 

.90 

.83 

.91 

.85 

.92 

.87 

.93 

.81 

.90 

.82 

.91 

.84 

.92 

.87 

.93 

6 

.84 

.91 

.86 

.93 

.89 

.94 

.90 

.95 

.85 

.92 

.87 

.93 

.89 

.94 

.92 

.96 

7 

.87 

.93 

.89 

.94 

.91 

.96 

.93 

.97 

.88 

.94 

.90 

.95 

.92 

.96 

.95 

.97 

8 

.89 

.94 

.91 

.95 

.94 

.97 

.96 

.98 

.91 

.95 

.92 

.96 

.95 

.97 

.98 

.99 

9 

.90 

.95 

.93 

.96 

.95 

.98 

.97 

.99 

.92 

.96 

.94 

.97 

.97 

.98 

1.00 

1.00 

10 

.92 

.96 

.94 

.97 

.97 

.98 

.99 

.99 

.94 

.97 

.96 

.98 

.99 

.99 

11 

.93 

.96 

.95 

.98 

.98 

.99 

1.00 

1.00 

.96 

.98 

.98 

.99 

1.00 

1.00 

12 

.94 

.97 

.96 

.98 

.99 

1.00 

.97 

.99 

.99 

1.00 

13 

.95 

.97 

.97 

.99 

1.00 

1.00 

.98 

.99 

1.00 

1.00 

14 

.96 

.97 

.98 

.99 

.99 

1.00 

15 

.96 

.98 

.99 

1.00 

1.00 

1.00 

16 

.97 

.98 

1.00 

1.00 

17 

.97 

.98 

1.00 

1.00 

18 

.98 

.99 

19 

.98 

.99 

20 

.98 

.99 

21 

.99 

.99 

22 

.99 

1.00 

23 

.99 

1.00 

24 

1.00 

1.00 

25 

1.00 

1.00 

RELATIVE  WEIGHTS  TO  TRANSITS  DEPENDING  ON  THE  STAR'S  DECLINATION. 

The  following  tables  of  the  probable  error  (e)  of  an  observation  of  a  transit  of  a  star  over  a 
single  line  have  been  derived  from  a  discussion  of  1047  transits  taken  in  February  and  March, 
1869,  at  San  Francisco,  by  Assistant  G.  Davidson,  with  the  large  transit  C.  S.  No.  3  (aperture  2f 
inches,  magnifying  power  85);  and  875  transits  taken  about  the  same  time  at  Cambridge  by 
Assistant  A.  T.  Mosman,  including  some  observations  by  Subassistant  F.  Blake,  with  the  large 
transit  C.  S.  No.  5  (aperture  2|  inches,  magnifying  power  100).  For  the  discussion  of  obser- 
vations with  a  smaller  instrument,  330  transits  were  used,  taken  in  September,  October,  and 
November,  1871,  at  Cleveland,  Ohio;  and  585  transits,  taken  in  December  and  January,  1871-72, 
at  Falmouth,  Ky.,  by  Assistant  E.  Goodfellow,  with  a  meridian  telescope  C.  S.  No.  13  (aperture 
If  inches,  magnifying  power  about  70). 


Transit  No.  3 

•  Transit  No.  5 

Meridian  telescope 
No.  13 

Meridian  telescope 
No.  13 

» 

W 

S 

(•) 

3 

(«) 

a 

(«) 

0 

s 

o 

s 

o 

s 

0 

s 

87.2 

±0.74 

86.9 

±0.66 

81.9 

±0.62 

76.3 

±0.20 

86.6 

0.49 

80.0 

0.20 

76.9 

0.18 

68.2 

0.16 

83.0 

0.38 

76.3 

0.19 

67.4 

0.11 

55.8 

0.13 

81.0 

0.31 

72.6 

0.12 

62.0 

0.14 

48.4 

0.15 

68.4 

0.12 

68.8 

0.11 

55.8 

0.09 

23.2 

0.102 

62.9 

0.088 

3.2 

0.066 

44.8 

0.088 

20.4 

0.089 

48  6 

0  075 

29  7 

0  067 

170 

01  1  n 

28.5 

0.058 

0  7 

0  071 

6  1 

OAQA 

7.8 

0.060 

DETERMINATION   OF    TIME. 

These  tabular  values  are  fairly  represented  by  the  expressions 

Transit,  No.  3  0)  =  V(0.060)2+(0.036)2  tan2  d 


39 


Transit,  No.  5  (£)=V(0-066)2+(0.036)2  tan2  d 

Meridian  telescope,  No.  13  (£)=V(0.069)2+(0.078)2  tan2  3 
Meridian  telescope,  No.  13  (s)=V(0.087)2+(0.055)2  tan2  8 

Combining  these  expressions  for  the  larger  and  smaller  instruments,  we  obtain 
(e)  =  V(0.063)2+(0.036)2  tan  2  «J  and  (e)  =  V(0.080)3+  (0.063)2   tan  2  d 

respectively,1  from  which  the  following  tables  of  probable  errors  (s),  of  relative  weights  p, 
and  of  the  multipliers  -^Jp  for  the  conditional  equations,  have  been  computed: 

Table  of  weights  to  transits  depending  on  the  star's  declination. 


» 

For  large  portable  transits 

For  small  portable  transits 

(•) 

P 

•Jp 

w 

p 

VP 

0         / 

s 

s 

" 

0 

±0.06 

1 

1 

±0.08 

1 

1 

10 

.06 

1 

1 

.08 

0.98 

1 

20 

.06 

0.98 

1 

.08 

.92 

0.96 

30 

.07 

.91 

0.95 

.09 

.83 

.91 

40 

.07 

.82 

.90 

.10 

.70 

.83 

45 

.07 

.76 

.87 

.10 

.62 

.79 

50 

.08 

.69 

.83 

.11 

.53 

.73 

55 

.08 

.61 

.78 

.12 

.44 

.66 

60 

.09 

.51 

.71 

.14 

.34 

.59 

65 

.10 

.40 

.63 

.16 

.26 

.51 

70 

.12 

.29 

.54 

.19 

.18 

.42 

75 

.15 

.18 

.43 

.25 

.10 

.32 

80 

.21 

.09 

.30 

.37 

.05 

.22 

85 

.42 

.02 

.15 

.72 

.01 

.11 

d    Ursse  Minoris 

86  37 

0.61 

0.011 

0.103 

1.1 

0.006 

0.075 

51  Cephei 

87  12 

0.74 

0.007 

0.085 

1.3 

0.004 

0.062 

ft    Ursse  Minoris 

88  46 

1.7 

0.001 

0.037 

2.9 

0.001 

0.027 

A     Ursse  Minoris 

88  59 

2.0 

0.001 

0.031 

3.5 

0.001 

0.023 

COMPUTATION  OF  AT  AND  a  BY  LEAST  SQUARES. 

A  field  computation  made  by  the  approximate  method  indicated  on  page  34  gives  values 
for  d  T,  a,  and  c,  which  are  of  a  high  degree  of  accuracy.  It  should  be  noted  that  the  derived 
values  of  a  and  c  depend  upon  all  the  observations  and  not  simply  upon  observations  on  a  few 
stars  only  of  the  set,  as  is  frequently  the  case  with  other  approximate  methods.  Experience 
shows  that  the  value  of  c  especially,  as  thus  derived  in  the  field  computation,  is  so  accurate 
that  a  value  derived  from  a  subsequent  rigid  least  square  adjustment  will  in  general  be  sub- 
stantially identical  with  it,  provided  the  stars  of  the  set  are  chosen  as  indicated  on  pages  34  and 
43.  Accordingly,  in  the  final  computations  by  this  method,  only  the  unknowns  aw,  aE,  and 
A  T  are  to  be  determined  by  least  squares,  while  c  is  taken  from  the  field  computations,  revised 
and  corrected  if  necessary.  This  method  of  computation  is  shown  below. 

Let  Ate  =  (a  —  t}  —  Cc  in  which  t  is  the  chronometer  time  of  transit  across  the  mean  line  of 
the  diaphragm  corrected  for  rate,  diurnal  aberration  and  inclination  and  ct  —  t  is  therefore  the 

1  The  following  formula  has  been  published  by  Dr.  Albrecht  on  p.  23  of  his  Formeln  und  Hiilfstafeln,  etc.,  Leipzig,  1894,  viz: 


d)=y  (0.05)«+  sec*  ) 

Putting  v=  85  for  the  magnifying  power  and  changing  sec  into  tan,  this  expression  is  equivalent  to 

(e)—  V(0-062)»+(0.037)s  tan"  ) 


40 


U.   S.   COAST   AND   GEODETIC    SURVEY    SPECIAL   PUBLICATION    NO.    14. 


quantity  on  the  last  line  of  the  field  record  and  computation  as  shown  on  pages  30-31.  Let  At 
be  an  assumed  value  of  the  chronometer  correction  and  dt  a  correction  to  At  to  be  derived  from 
the  computation.  The  final  value  of  the  chronometer  correction  will  then  be  AT=At  +  dt. 
Let  d,  for  each  star=Jic  — At. 

Then  for  each  star  observed  an  observation  equation  of  the  form 

Vp  St  +  -JpAa  =  V?  d, 

may  be  written,  in  which  the  weights  p  are  assigned  according  to  the  tables  on  pages  38-39. 

In  forming  the  normal  equations  each  half  set,  made  with  the  horizontal  axis  in  one  posi- 
tion, is  treated  independently  of  the  other  half  set. 

The  normal  equations  corresponding  to  the  half  set  made  with  illumination  (or  bright 
band)  to  the  westward  are 

Ipdt   +  IpAaw  =  Ipd 
IpAdt  +  IpA^  =  Ip  Ad 

and  similarly  for  the  other  half  set. 

The  most  convenient  arrangement  of  this  computation  is  shown  below,  this  example  being 
a  computation  of  the  time  set  treated  on  pages  29-31  and  34. 


WASHINGTON,  D.  C.,  May  17,  1896. 

c=+.032 


J(=-4S.01 


Star 

Band 

tt-( 

C 

Cc 

J(c 

d 

A 

P* 

pA 

pA* 

pd 

pAd 

Aa 

AT 

J 

pA 

pffi 

17  H.Can.Ven. 

W 

-4.07 

+1.26 

+.04 

-4.11 

-  .10 

+  .02 

.83 

+.02. 

.  00 

-.08 

.00 

+  .01 

-4.12 

+  .10 

+.08 

.0083 

13    Urs.  Maj. 

W 

-4.09 

+  1.55 

+  .05 

-4.14 

-  .13 

-  .30 

.69 

-.21 

.06 

-.09 

+.03 

-  .18 

-3.96 

-.06 

-.04 

25 

jj    Bootis 

W 

-3.69 

+1.06 

+  .03 

-3.72 

+  .29 

+  .36 

.98 

+.35 

.13 

+  .28 

+.10 

+  .22 

-3.94 

-.08 

-.08 

63 

II  Bootis 

W 

-3.89 

+1.13 

+.04 

-3.93 

+  .08 

+  .22 

.93 

+.20 

.04 

+  .07 

+.02 

+  .13 

-4.06 

+  .04 

+  .04 

15 

a   Draconis 

W 

-4.52 

+2.36 

+  .08 

-4.60 

-  .59 

-1.03 

.40 

-.41 

.42 

-.24 

+.24 

-  .62 

-3.98 

-.04 

-.02 

06 

3.83 

-.05 

.65 

-.06 

+  .39 

04 

d  Bootis 

E 

-3.94 

-1.11 

-.04 

-3.90 

+  .11 

+  .25 

.93 

+  .23 

.06 

+  .10 

+.03 

+  .14 

-4.04 

+.02 

+  .02 

a  Bootis 

E 

-3.81 

-1.06 

-.03 

-3.78 

+  .23 

+  .35 

.98 

+  .34 

.12 

+  .23 

+  .08 

+  .19 

-3.97 

-.05 

-.05 

25 

A  Bootis 

E 

-4.23 

-1.46 

-.05 

-4.18 

-  .17 

-  .19 

.74 

-.14 

.03 

-.13 

+  .02 

-  .10 

-4.08 

+  .06 

+  .04 

27 

e  Bootis 

E 

-4.29 

-1.64 

-.05 

-4.24 

-  .23 

-  .38 

.65 

-.25 

.09 

-.15 

+  .06 

-  .21 

-4.03 

+  .01 

+  .01 

01 

5  Urs.  Min. 

E 

-5.44 

-4.18 

-.13 

-5.31 

-1.30 

-2.53 

.16 

-.40 

1.02 

-.21 

+  .53 

-1.37 

-3.94 

-.08 

-.01 

10 

3.46 

-.22 

1.32 

-.16 

+.72 

.0259 

*  These  weights  are  taken  from  the  column  headed  "  For  large  portable  transits  "  in  the  table  on  p.  39. 

Normal  equations: 

+3.83  d  t-.Oo  aw=-  .06 
-    05d*+.65aw=  +  .39 
aw=+».601 


+3.46  St-   .22aE=-  .16 
-  .22  St+1.32  aE=  +  .72 
aE=+«.543 
3t=-'.012 


+7.29  Q-  .27  7=1 
-  .27  Q+1.97  q=0 


At  14h  02m  JT=-43.020 
Q=0.138 


£[=±'.044 
£  =-!-s.016 


In  the  above  computation  a  check  on  the  correctness  of  the  assumed  value  of  c  is  furnished 
by  the  nearness  of  agreement  of  the  two  values  of  dt  resulting  from  the  two  groups  of  stars. 
The  normal  equations  are  solved  most  conveniently  by  successive  approximations,  as,  for 


DETERMINATION   OF    TIME.  41 

instance,  in  the  second  equation  the  value  of  aw  can  be  closely  derived  at  once  on  the  assumption 
that  dt  is  small.  The  residuals  (J)  are  taken  for  each  group  separately,  using  its  own  dt1  to 
derive  a  A  T  for  this  purpose,  and  the  sums  of  the  pJ's  should  of  course  nearly  equal  zero  for 
each  set.  The  probable  error  of  a  single  observation  of  unit  weight  is 

.,  =  0.674^1 


^^ 

\  n0  - 


where  2pJ2  is  the  sum  of  the  weighted  squares  of  the  residuals  (last  column  in  form),  n0  is  the 
number  of  stars  and  ne  is  the  number  of  unknown  quantities  or  number  of  normal  equations, 
remembering  in  this  example  that  there  are  four  unknowns,  dt,  aw,  aE,  and  c,  the  latter  being 
taken  from  the  field  computation.  To  obtain  the  probable  error  £  of  the  computed  AT,  add 
the  corresponding  normal  equations  of  the  two  sets,  put  Q  in  place  of  dt,  g  in  place  of  a,  1  in 
place  of  2pd,  and  0  in  place  of  2pAd,  as  shown.  Then  £  =  e^Q. 

THE  COMPLETE  LEAST  SQUARE  COMPUTATION. 

When  time  observations  are  taken  in  Alaska  unusual  conditions  are  encountered,  arising 
from  the  high  latitude  of  the  station  —  from  55°  to  65°  for  the  regions  in  which  the  Survey 
observers  are  called  upon  to  observe  most  frequently.  Zenith  stars  are  there  slow-moving  stars 
(and  consequently  have  small  weights)  ;  for  stars  between  the  zenith  and  the  pole  pA  is  com- 
paratively small;  the  rapidly  moving  stars  are  far  to  the  southward  of  the  zenith,  and  it  is  easy 
to  observe  subpolars,  as  the  northern  horizon  is  far  below  the  pole.  Moreover  the  very  prevalent 
cloudy  weather  is  apt  to  break  in  .  upon  any  previously  arranged  program.  The  combined 
result  of  these  conditions  is  in  general  that  the  sets  of  stars  actually  observed  are  poorly  balanced; 
that  is,  the  algebraic  sum  of  the  A  factors  for  each  half  set  and  of  the  C  factors  for  the  whole 
set  will  differ  considerably  from  zero.  In  extreme  cases  it  is  sometimes  desirable  to  resort 
to  the  complete  least  square  computation  in  which  c,  aw,  aE,  and  AT  are  all  derived  by  the 
principle  of  least  squares. 

We  here  start  with  a  —  t  (as  shown  on  pp.  30-31),  and  the  remaining  notation  stands  as  on 
page  40,  except  that  we  must  here  distinguish  by  the  subscripts  w  and  E  between  A  factors  belong- 
ing to  the  two  half  sets. 

An  observation  equation  of  one  of  the  following  forms  may  be  written  for  each  star  observed: 

•Jpdt  +  -JpA£aE 

-Jpdt 

The  normal  equations  will  be— 


IpCdt  +  IpAECaE 

The  following  will  serve  as  a  concrete  illustration  of  this  method  of  computation.  The  only 
preliminary  assumption  in  this  computation  is  an  approximate  value  of  the  chronometer  correc- 
tion, At. 

Owing  to  the  high  latitude  of  St.  Michael,  63°  29',  the  time  stars  are  all  south  of  the 
zenith,  and  the  average  value  of  A  is  far  from  zero. 

1  Tile  two  3t's  here  happen  to  be  so  nearly  equal  that  J's  are  the  same  as  if  taken  by  using  the  J  T  for  the  whole  group. 


42  U.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.   14. 

ST.  MICHAEL,  ALASKA,   March  19,  1891. 

J<= -20.10. 


Star 

Clamp 

a-t 

d 

A 

C 

P 

pA 

pC 

pA* 

pAC 

pC' 

pi 

pAd 

pCd 

Aa 

Cc 

AT 

J 

pA 

T>f 

s 

s 

1 

E 

-21.27 

-1.17 

+  .66 

-1.13 

0.9 

+  .59 

-1.02 

.39 

-  .67 

1.15 

-1.05 

-  .69 

+  1.-19 

—  .89 

-  .21 

-20.17 

+  .05 

+.04 

.0022 

2 

E 

-21.22 

—  1.12 

+  .72 

-1.08 

0.9 

+  .65 

-  .97 

.47 

-  .70 

1.05 

-1.01 

-  .73 

+  1.09 

-  .97 

-  .20 

-20.  05 

-.07 

-.06 

44 

3 

E 

-21.40 

-1.30 

+  .76 

-1.05 

0.9 

+  .68 

-  .94 

.52 

-  .72 

.99 

-1.17 

-  .89 

+  1.23 

-1.02 

-  .19 

-20.  19 

+  .07 

+  .06 

44 

4 

E 

-23.09 

-2.99 

+2.89 

+4.58 

0.08 

+  .23 

+  .37 

.66 

+1.06 

1.68 

-  .24 

-  .69 

-1.10 

-3.88 

+  .84 

-20.05 

-.07 

-.01 

04 

5 

E 

-21.23 

-1.13 

+  .73 

-1.07 

0.9 

+  .66 

-  .96 

.48 

-  .70 

1.03 

-1.02 

-  .74 

+  1.09 

-  .98 

-  .20 

-20.05 

-.07 

-.06 

44 

+2.81 

2.52 

-1.73 

-3.74 

01 

6 

W      j-20.98 

-0.88 

+  .85 

+1.01 

1.0 

+  .85 

+  1.01 

.72 

+  .86 

1.02-  .88 

-  .75 

-  .89 

-1.05 

+  .18 

-20.11 

-.01 

-.01 

7 

W 

-20.86 

-0.76 

+  .72 

+  1.08 

0.9 

+  .65+  .97 

.47 

+  .70 

1.08 

-  .  68  -  .49 

-  .73 

-  .89 

+  .20 

—20.17 

+.05 

+.04 

22 

8 

W 

-20.70 

-0.60 

+  .64 

+  1.14 

0.9 

+  .58+1.03 

.37 

+  .66 

1.17 

-  .54 

-  .35 

-  .62 

-  .79 

+  .21 

-20.12     .00 

.00 

00 

9 

W 

-20.95 

-0.85 

+  .85 

+  1.01 

1.0 

+  .85 

+  1.01 

.72 

+  .86 

1.02 

-.85 

-  .72 

-  .86 

-1.05 

+  .18 

-20.08 

-.04 

-.04 

16 

10 

W 

-25.39 

-5.29 

+3.46 

-5.83 

0.05 

+  .17 

-  .29 

.60 

-1.01 

1.70 

-  .26 

-  .92 

+  1.54 

-4.27 

-1.07 

-20.05 

-.07 

.00 

02 

7.53 

+3.10 

+0.21 

2.88 

+2.07 

11.86 

-7.70 

-3.23 

+  1.94 

0199 

Normal  equations: 


+7.  53  St  +2.  81  aE+3. 10  aw+    0.  21  c    =-7.  70 
+2.  81  St  +2.  52  <z.E  1.  73  c    =  -3.  74 

+3. 10  St  +2.  88  aw+    2.  07  c   =  -3. 23 

+0. 21  St  -1.  73  aE+2.  07  aw+  11.  86  c    =+1.  94 

c   =+0.183 

aE=- 1.342 

aw=- 1.233 

ot  =-0-02 
At  S.h5  AT  =  -20.12 
Q     =  .79 

e  =±.  035 

The  remarkably  large  value  for  Q  arises  from  the  fact  that  the  azimuth  errors,  aw  and  aE 
are  but  feebly  determined,  see  column  headed  pA  and  the  normal  equations. 

Sometimes  it  is  assumed  that  the  azimuth  error  is  the  same  for  both  halves  of  a  set,  and 
the  distinction  between  aw  and  aE  is  dropped  and  a  single  a  derived  from  the  whole  set,  the 
normal  equations  being  modified  accordingly.  This  procedure  is  entirely  justifiable  if  the 
azimuth  error  during  the  two  half  sets  is  actually  the  same.  If  the  two  azimuths  really  differ, 
some  error  will  be  introduced  into  the  computed  results  by  this  procedure,  and  the  error  so 
introduced  will  be  larger  the  greater  is  said  difference.  Experience  shows  that  the  instability 
of  the  instrument  in  azimuth  is  in  general  sufficient  to  make  it  desirable  to  distinguish  between 
the  two  azimuth  errors  if  accurate  results  are  desired,  except  when  there  are  but  few  stars 
observed  in  the  set,  say,  seven  or  less. 

THE   SELECTION   OF   STARS. 

The  stars  shown  in  the  observing  list  (p. 18)  and  used  in  the  computation  on  pages  21,22  and 
26  were  chosen  by  the  method  now  used  for  longitude  work  in  latitudes  less  than  50°.  In  each 
half  set  there  are  five  to  seven  time  stars  (six  stars  preferred),  a  time  star  being  one  which  has 
an  A  factor  less  than  unity.  These  stars  are  so  selected  that  the  algebraic  sum  of  the  A  factors 
in  a  half  set  shall  not  be  greater  than  unity.  It  is  desirable  to  have  the  algebraic  sum  of  the 
A  factors  of  the  stars  in  a  half  set  as  small  as  can  be  obtained  by  the  use  of  good  judgment 
in  their  selection,  but  it  is  not  desirable  to  reduce  the  number  of  stars  per  hour  to  be  observed 
in  order  to  improve  the  balancing  of  the  A  factors,  if  the  balancing  is  already  within  the 
specified  limit. 

In  endeavoring  to  obtain  the  maximum  number  of  stars  per  hour,  subject  to  the  condition 
of  the  balancing  of  the  A  factors,  consideration  must  be  given  the  question  of  level  readings 


DETERMINATION   OF   TIME.  43 

and  reversals  of  the  instrument.  Ample  time  should  be  provided  for  the  performance  of  these 
operations.  In  longitude  work  allowance  must  be  made  for  the  exchange  of  time  signals, 
which,  if  the  stations  are  not  very  far  apart,  usually  takes  place  between  the  two  sets — that 
is,  between  the  second  and  third  half  sets.  The  exchange  may  be  made,  however,  at  any  time 
during  the  observing  period  if  there  is  trouble  in  getting  a  clear  wire  between  the  two  observa- 
tories or  if  clouds  break  up  prearranged  sets  of  stars.  An  observer  soon  learns  from  practice 
how  much  time  must  be  allowed  for  the  different  operations. 

It  is  desirable,  but  not  necessary,  to  observe  the  same  stars  at  both  stations  when  deter- 
mining a  difference  of  longitude.  This  is  of  less  importance,  however,  than  securing  rapid 
observations  with  the  A  factors  in  each  half  set  well  balanced.  When  the  two  stations  are  not 
distant,  many  of  the  stars  observed  at  one  station  will  necessarily  be  observed  at  the  other. 

In  longitude  work  the  observations  each  night  consist  normally  of  four  half  sets  of  six 
stars  each,  with  a  reversal  of  the  instrument  between  each  two  consecutive  half  sets.  The 
reversal  of  the  instrument  after  each  of  the  half  sets  is  a  precaution  which  experience  has 
justified,  for  should  only  three  half  sets  be  observed  (through  interference  of  clouds  or  for  other 
reasons)  two  sets  can  still  be  obtained  by  combining  the  first  and  second  and  the  second  and 
third  half  sets,  thus  obtaining  two  corrections  to  the  chronometer  and  its  rate. 

Where  it  is  desired  to  use  the  azimuth  star  method  of  solution  shown  on  pages  34  and  40,  a  dif- 
ferent selection  of  stars  is  to  be  made.  A  half  set  consists  of  five  stars  following  each  other  in  rapid 
succession,  so  chosen  that  the  algebraic  sum  of  the  A  factors  of  the  four  time  stars  (each  near 
the  zenith)  will  be  nearly  zero,  and  that  the  azimuth  star  of  each  half  set  will  have  its  A  factor 
greater  than  unity,  and  yet  not  be  so  near  the  pole  as  to  render  the  star's  transit  across  the 
field  of  observation  so  slow  as  to  produce  long  waits  between  observations.  In  a  time  set, 
chosen  as  above,  observation  upon  the  azimuth  star  in  each  half  set  serves  principally  to 
determine  the  azimuth  error  of  the  instrument,  but  has  little  effect  upon  the  computed  time, 
since  this  is  almost  independent  of  the  azimuth  error  (the  sum  of  the  A  factors  of  the  time 
stars  being  nearly  zero  for  each  half  set).  Where  only  approximate  time  is  required,  the 
number  of  time  stars  in  a  half  set  may  be  reduced  to  two,  one  north  and  one  south  of  the  zenith. 

In  high  latitudes  (more  than  about  50°),  it  is  not  feasible  to  secure  time  sets  with  well- 
balanced  A  factors,  since  the  stars  between  the  zenith  and  the  pole  have  comparatively  small 
A  factors,  which  become  relatively  still  smaller  after  weights  are  assigned.  This  condition 
prevents  any  but  a  comparatively  weak  determination  of  the  azimuth  error  of  the 'instrument. 
In  such  latitudes  it  is  therefore  desirable  to  select  sets  of  stars  which  will  be  solved  by  rigid 
least-square  methods.  Under  normal  conditions  there  should  be  six  stars  in  each  half  set, 
and  while  the  algebraic  sum  of  the  A  factors  in  each  half  set  should  be  kept  as  small  as  can  be 
conveniently  done,  no  very  slow-moving  stars  should  be  introduced  for  this  purpose.  One 
azimuth  star  with  a  declination  between  55°  and  75°  should  be  selected  and  observed  below 
the  pole. 

The  preliminary  or  field  computations  may  be  made  like  that  shown  on  page  26.  The 
final  least  square  computations  are  made  at  the  office. 

As  has  already  been  stated  (p.  25),  the  preference  is  now  given  to  the  American  Ephemeris 
over  other  star  lists,  as  it  contains  the  apparent  places  of  more  stars  than  other  available  cata- 
logues. It  is  well  to  obtain  all  stars,  when  possible,  from  a  single  catalogue,  but  this  is  not 
essential.  It  may  be  considered  as  almost  essential,  certainly  so  from  an  economic  standpoint, 
to  use  only  stars  for  which  apparent  places  are  published.  The  time  and  labor  consumed  in 
computing  the  apparent  right  ascension  of  stars  for  which  only  mean  places  are  available 
add  to  the  cost  of  both  the  field  and  office  work.  Furthermore,  it  will  be  found  that  sufficient 
stars  can  be  selected  for  all  time  work  in  the  northern  hemisphere  from  such  catalogues  as  the 
American  Ephemeris  and  Nautical  Almanac  or  the  Berliner  Astronomisches  Jahrbuch,  and  the 
selection  of  mean  place  stars  is  unnecessary. 

DETERMINATION   OF  EQUATORIAL   INTERVALS. 

The  equatorial  intervals  of  the  lines  of  the  diaphragm  are  needed  to  reduce  incomplete 
transits.  (See  p.  32.) 


44 


U.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.   14. 


To  determine  these,  select  complete  transits  of  stars  of  large  declination. 

Let  tlt  i2,  t3  ......  in  be  the  observed  times  of  transit  over  the  successive  lines,  tm,  their 

mean,  and  -iu  i2,  i,  ......  in  their  equatorial  intervals  from  the  mean  line  and  d  the  declination 

of  the  star: 


\  =  (t1—  tm)  cos  d 
i2=(t2-tm)  cos  d 
etc. 

"in  =  (<n  —  <m)  COS  d 

also          0  =  i1+i2+'i3  ......  +in. 

The  intervals  of  the  lines  j  eas  ,  i  of  the  mean  line  will  then  be  |      >  at  upper  culmination 

For  stars  witlu'n  10°  of  the  pole  (as  for  d  Urs.  Min.,  51  Cephei,  Polaris,  and  A  Urs.  Min.) 
use  the  formulae: 

ij  =  (<,  —  tm)  cos  d  -/  cos  TJ 
etc.          ^___ 

^n  =  (<n  -  tm)  COS  d  $  COS   Tn 

where  TU  T2,  TS  ......  rn  are  the  hour  angles  of  the  circumpolar  star  for  the  successive  lines. 

When  it  is  necessary  to  use  the  more  exact  formula  for  circumpolars  as  given  above,  the 
table  on  page  32  will  be  found  convenient. 

If  the  chronometer  rate  exceeds  15s  per  day  it  will  be  desirable  to  take  it  into  account  in 
computing  the  equatorial  intervals. 

A  convenient  form  for  the  computation  of  equatorial  intervals  follows.  The  observations 
used  were  made  by  Assistant  Fremont  Morse  at  Sitka,  Alaska,  in  1894,  with  Meridian  Telescope 
No.  7,  and  by  the  eye  and  ear  method. 

K  Draconis.     3=70°  22'  27".     Log.  cos  3=9.52618.     Clamp  West. 


Line 

May  14 

May  15 

May  16 

May  IS 

Mean 

Log.  mean 

Log.  i 

(equatorial 

interval) 

S 

S 

S 

S 

S 

S 

1 

^1  —  ^m 

-87.  60 

-88.00 

-87.  10 

-87.60 

-87.  575 

1.  94238 

1.  46856 

-29.  414 

2 

tz  —  tm 

-44.  60 

-44.00 

-44.  60 

-44.  60 

-44.  450 

1.  64787 

1.  17405 

-14.  930 

3 

t3  —  tm 

-  0.10 

0.00 

+  0.40 

+  0.40 

+  0.  175 

9.  24304 

8.  76922 

+  0.  059 

4 

ti  —  t,a 

+43.  90 

+44.00 

+43.  90 

+43.  40 

+43.  800 

1.  64147 

1.  16765 

+14.  711 

5 

<5-<m 

+88.  40 

+88.00 

+87.  40 

+88.  40 

+88.  050 

1.  94473 

1.  47091 

+29.  574 

The  quantities  (^-<m),  (t2-tm),  etc.,  for  each  date  were  taken  directly  from  the  record  of 
observations. 

The  equatorial  intervals  were  thus  computed  from  observations  upon  three  different  stars 
and  the  means  taken. 

It  is  not  necessary  to  make  special  observations  to  determine  the  equatorial  intervals. 
Complete  transits  observed  during  the  regular  progress  of  time  observations  may  be  utilized 
for  that  purpose.  If  observations  upon  stars  of  large  declination  are  not  available,  observa- 
tions upon  stars  of  small  declination  may  be  used,  and  will  be  found  to  give  almost  as  accurate 
values  for  the  equatorial  intervals. 

When  pressed  for  time  in  the  field  an  incomplete  transit  of  a  star  may  be  reduced  by  assuming 
that  actual  intervals  between  lines  on  that  star  are  the  same  as  on  some  preceding  date  on 
which  a  complete  transit  of  that  star  was  observed  at  that  station.  The  formulse  on  page  32 
may  then  be  used  by  dropping  the  factor  sec  d  and  substituting  actual  intervals  for  equatorial 
intervals. 

PIVOT  INEQUALITY. 

The  pivot  inequality  should  be  determined  with  the  instrument  mounted  upon  a  very 
stable  pier  in  a  room  in  wlu'ch  the  rate  of  change  of  temperature  is  small  during  the  observa- 
tions. The  observations  consist  of  a  series  of  readings  of  the  striding  level  as  indicated  in  the 


DETERMINATION   OF   TIME. 


45 


example  of  record  and  computation  given  below.  The  notation  is  the  same  as  on  pages  22-23; 
that  is,  /?,„  and  /?e  indicate  the  apparent  inclination  of  the  telescope  axis  in  each  of  its  two  posi- 
tions as  given  directly  by  the  readings  of  the  striding  level.  Then  the  pivot  inequality 


and  is  to  be  expressed  in  seconds  of  time. 

Observations  for  inequality  of  pivots  of  transit,  No.  19. 


[Station,  Atlanta,  Ga.,  MaA  12,  18%.    G.  R.  P.,  observer.] 


"" 

Band  west 

Band  east 

Object  glass  south 

Object  glass  north 

Zenith 
distance 

Time 

Temper- 
ature 

Zw-2t 

Sw  —  Ze 

0e  —  @u> 

4 

Level 

4 

=  $W 

Level 

4 

=  Pc 

=  P 

W.  end 

E.end 

W.end 

E.end 

0 

h  m 

Of 

div 

div 

div 

div 

div 

div 

div 

38 

9  43  a.  m. 

33 

33.5 

22.0 

33.4 

21.7 

20.8 

34.8 

-  .625 

21.0 

34.0 

-.325 

+.075 

43 

20.4 

33.9 

21.0 

34.0 

32.4 

21.9 

-  .750 

33.1 

21.8 

-.425 

+.081 

48 

20.2 

33.9 

20.3 

33.4 

32.2 

21.9 

-  .850 

32.  1 

21.8 

-.700 

+.038 

43 

31.8 

21.9 

32.7 

21.1 

19.7 

33.9 

-1.075 

20.1 

33.3 

-.400 

+.  169 

38 

10  03  a.  m. 

35 

19.7 

33.8 

20.1 

33.1 

31.9 

21.3 

-  .875 

32.0 

21.1 

-.525 

+.088 

Mean,    band    west,    object   glass 
south,   and   band   east,   object 

+.090 

glass 

north 

Band  west 

Rand  east 

Object  glass  north 

Object  glass  south 

Zenith 
distance 

Time 

Temper- 
ature 

sw-le 

Sw-Se 

Pl-Pw 

4 

Level 

4 
=ffu> 

Level 

4 
—t» 

=  P 

W.end 

E.end 

W.end 

E.end 

o 

h    m 

op 

div 

div 

div 

div 

div 

div 

div 

38 

10  07  a.  m. 

35 

19.7 

33.1 

19.4 

33.6 

31.9 

20.9 

-.600 

31.9 

20.9 

-.800 

-.050 

43 

31.9 

20  9 

31.7 

20.9 

19.1 

33.3 

-.800 

19.1 

33.2 

-.825 

-.006 

48 

19.3 

33.0 

19.1 

33.3 

31.5 

20.9 

-.775 

31.7 

20.9 

-.850 

-.019 

43 

31.3 

20.9 

31.1 

21.0 

19.0 

33.2 

-.950 

18.9 

33.2 

-1.050 

-.025 

33 

10  27  a.  m. 

36 

19.0 

33.1 

18.8 

33.7 

31.7 

20.5 

-.725 

31.2 

20.9 

-1.  150 

-.106 

Mean,    band    west,    object  glass 

-.041 

north,   and   band   east,   object 

glass  south 

Mean,    band    west,    object   glass 

+.090 

south,   and   band   east,   object 

glass  north 

Mean 

+.024 

1  division  of  striding  level=l//.850=OM23 

p=  +  .024  div.=OM23X.024=+0.003  sec- 

ond of  time 

46  U.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.   14. 

In  determining  the  pivot  inequality  the  level  readings  are  made  as  in  observing  time, 
reversing  the  telescope  between  the  readings.  Observations  should  be  made  in  two  groups, 
reversing  the  relation  between  the  positions  of  the  band  and  object  glass  as  shown  in  the  example. 
This  is  done  to  partially  eliminate  the  effect  of  the  pivots  not  being  truly  circular  in  cross  section. 
In  the  example  shown  there  is  a  systematic  though  unimportant  difference  in  p  for  the  two 
positions  A  complete  investigation  of  the  pivots  would  involve  level  readings  at  all  angles 
from  the  zenith,  from  0°  to  90°,  but  the  ordinary  form  of  level  will  not  permit  readings  closer 
than  30°  or  40°,  and  stars  are  not  often  observed  more  than  50°  from  the  zenith.  In  the  example; 
given  the  observations  were  from  38°  to  48°  zenith  distance,  less  weight  being  given  to  the  latter 
angle  at  which  few  star  observations  are  made. 

A  less  satisfactory  value  for  the  pivot  inequality  may  be  obtained  from  the  level  readings 
made  in  connection  with  the  time  observations. 

Since  the  correction  for  pivot  inequality  has  opposite  signs  for  the  two  halves  of  a  time  set, 
its  effect  on  the  determined  clock  correction  is  very  small  for  a  set  which  has  the  same  number 
of  stars  in  each  half.  The  question  of  when  the  pivot  inequality  correction  is  to  be  applied 
and  when  not,  should  be  decided  after  a  consideration  of  the  absolute  value  of  the  correction 
but  the  difference  in  the  sums  of  the  B  factors  for  the  two  half  sets  should  also  be  considered. 
Most  of  the  instruments  used  at  present  in  this  Survey  have  had  their  pivots  refinished  and  their 
pivot  inequality  made  practically  zero.  With  these  instruments  it  is  not  usually  necessary 
to  consider  this  correction  when  making  the  computations  for  time. 

DETERMINATION  OF  LEVEL  VALUE. 

The  most  accurate  way  of  determining  the  value  of  one  division  of  a  level  is  by  means  of 
a  level-trier,  wliich  consists  of  a  bar  the  support  of  which  at  one  end  is  a  micrometer  screw. 
The  level  tube  to  be  tested  is  placed  on  this  bar.  The  method  of  observing  and  computing  is 
shown  in  the  following  example.  In  the  level-trier  used  one  division  of  the  micrometer  head 
equals  one  second  of  arc;  that  is,  a  movement  of  one  division  changes  the  angular  position  of 
the  bar  by  one  second.  The  first  part  of  these  observations  was  simply  for  the  purpose  of  test- 
ing the  uniformity  of  the  tube,  changing  the  angle  by  5"  intervals.  In  determining  the  level 
value  about  the  same  length  of  bubble  is  employed  that  is  used  in  the  field  observations. 


DETERMINATION   OF   TIME. 


47 


Determination  of  value  of  one  division  of  stride  level  of  meridian  telescope  No.  9.  Chamber  vial 
175  mm.  by  15  mm.,  marked  7526,  2" .02  K.  and  E.,  mounted  by  springs.  Length  of  bubble 
used,  35  div.  =  70  mm.  E.  G.  F.,  observer.  Mean  temperature,  12°. 3  C. 


Chamber  left 

Chamber  right 

Bubble  reading 

Movement 

Bubble  reading 

Movement 

Level- 
trier 
reading 

Value  of 
one  divi- 
sion of 
level 

Level- 
trier 
reading 

Value  of 
one  divi- 
sion of 
level 

Left 
end 

TUsht 
end 

Level- 
trier 

Bubble. 
Mean  of 
two  ends 

Left 
end 

Right 
end 

Level- 
trier 

Bubble. 
Mean  of 
two  ends 

// 

div 

div 

// 

div 

// 

// 

div 

div 

// 

div 

// 

25 

-0.1 

35.2 

75 

GO.  4 

25.8 

30 
35 
40 
45 
50 

2.4 
4.9 
7.4 
10.1 
12.7 

37.7 
40.2 
42.7 
45.4 
48.0 

5 
5 
5 
5 

2.5 

2.5 
2.7 
2.6 

80 
85 
90 
95 
100 

57.7 
55.3 
52.9 
50.2 
47.5 

23.  1 
20.7 
18.3 
15.6 
12.9 

5 
5 
5 
5 

2.4 
2.4 
2.7 
2.7 

55 

15.3 

50.6 

* 

O    ft 

105 

44.9 

10.3 

K 

o    7 

60 
65 
70 

75 

17.9 
20.3 
22.9 
25.5 

53.2 
55.  6 
58.2 
60.8 

5 
5 
5 

2.4 
2.6 
2.6 

110 
115 
120 
125 

42.  2 
39^6 
37.0 
34.5 

7.6 
5.0 
2.4 
-0.1 

5 
5 
5 

2.6 
2.6 
2.5 

25 

75 

-0.2 
25.5 

35.0 
60.7 

50 

25.7 

1.945 

75 
125 

60.9 
34.6 

26.3 
0.0 

£0 

26.3 

1.901 

35 
65 

4.7 
20.5 

39.9- 
55.7 

30 

15.8 

1.899 

85 
115 

55.9 
39.8 

21.  2            „ 
5.1 

16.1 

1.863 

40 

CO 

7.4 
17.9 

42  6 
53.1 

20 

10.5 

1.905 

90 
110 

53.2 
42.4 

18.5 

7.  7 

20 

10.8 

1.  852 

45 

55 

10.1 
15.4 

45.3 
50.6 

10 

5.3 

1.887 

95 
105 

50.4 
44.9 

15.8 
10.3 

10 

5.5 

1.818 

Mean 

,  chamber  left 

1  909 

!    Mean 

chamb 

?r  ric;ht 

1.  859 

Final  mean 

1  div 

=2  mm 

.  =  1/X.SJ 

!4  at  12° 

.30 

If  the  level  vial  is  so  held  in  its  metallic  mounting  that  there  is  any  possibility  that  it  may 
be  put  under  stress  by  a  change  of  temperature,  it  is  advisable  to  determine  the  value  of  a 
division  with  the  tube  in  its  mounting  at  two  or  more  widely  different  temperatures.  Level 
vials  are  now  usually  mounted  witli  springs,  so  as  to  avoid  such  stresses. 

If  an  observer  is  forced  to  determine  the  value  of  a  level  division  in  the  field,  remote  from 
a  level-trier — -after  some  accident,  for  example — he  must  devise  some  method  of  utilizing  what- 
ever apparatus  is  at  Ids  disposal  for  that  purpose. 

If  a  telescope  having  an  eyepiece  micrometer  fitted  for  measuring  altitudes  or  zenith  dis- 
tances is  available,  the  unknown  angular  value  of  a  level  division  may  be  found  by  comparison 
with  the  known  angular  value  of  a  division  of  the  micrometer.  Place  the  level  in  an  extempo- 
rized mounting  fixed  to  the  telescope  so  that  the  level  vial  is  parallel  to  the  plane  in  winch  the 
telescope  rotates  (about  its  horizontal  axis).  Point  with  the  micrometer  upon  some  distant 
well-defined  fixed  object  and  read  the  micrometer  and  level.  Change  the  micrometer  reading 
by  an  integral  number  of  divisions,  point  to  the  same  object  again  by  a  movement  of  the  tele- 
scope as  a  whole,  and  note  the  new  reading  of  the  level.  Every  repetition  of  tin's  process  gives 
a  determination  of  the  level  value  in  terms  of  the  micrometer  value. 

If  another  level  of  sufficient  sensibility  and  of  which  the  value  is  well  known  is  available, 
it  may  be  used  as  a  standard  with  which  to  compare  the  unknown  level.  Put  the  unknown 
level  in  an  extemporized  mounting,  fastened  to  that  of  the  known  level  in  such  a  way  that  the 
two  level  vials  are  parallel  or  nearly  so.  Adjust  so  that  both  bubbles  are  near  the  middle  at 
once.  Compare  corresponding  movements  of  the  two  bubbles  for  small  changes  of  inclination 
common  to  the  two  levels. 


48  U.  S.   COAST  AND  GEODETIC   SURVEY  SPECIAL  PUBLICATION   NO.   14. 

DISCUSSION  OF  ERRORS. 

The  various  errors  which  affect  the  final  result  of  any  astronomic  observation  may  be 
grouped  into  three  separate  classes  with  respect  to  their  sources,  and  consequently  the  pre- 
cautions which  must  be  taken  against  them  fall  under  the  same  general  heads.  They  are: 
(1)  External  errors,  or  errors  arising  from  conditions  outside  the  observer;  (2)  instrumental 
errors,  due  to  the  instrument,  and  arising  from  imperfect  construction 1  or  imperfect  condition 
of  the  instrument,  from  instability  of  the  relative  positions  of  the  different  parts,  etc.;  (3) 
observer's  errors,  due  directly  to  the  observer,  arising  from  liis  unavoidable  errors  of  judgment 
as  to  what  he  sees  and  hears  and  from  the  fact  that  nerves  and  brain  do  not  act  instantaneously. 
By  the  phrase  "Errors  of  observation"  is  meant  the  combined  errors  arising  from  all  these 
sources. 

The  principal  external  errors  in  transit  observations  for  time  arise  from  errors  in  the  assumed 
right  ascensions  of  the  stars  and  from  lateral  refraction  of  the  light  from  the  stars. 

If  the  right  ascensions  of  all  stars  observed  are  taken  from  the  American  Ephemeris  and 
Nautical  Almanac  or  the  Berliner  Astronomisches  Jahrbuch,  the  probable  error  of  a  right 
ascension  will  be  upon  an  average  about  ±0.S03,  except  for  stars  of  large  declination,  for  which 
this  estimate  must  be  increased.  The  right  ascensions  are  subject  also  to  small  constant  errors 
with  which  the  geodesist  is  hardly  concerned,  because  of  their  smallness  and  because  they  are 
almost  completely  eliminated  from  Ms  final  results.  When  the  same  stars  are  used  at  both 
stations  in  determining  a  difference  of  longitude  the  errors  of  the  right  ascensions  are  com- 
pletely eliminated  from  the  determined  difference  of  longitude. 

If  one  considers  how  small  are  the  lateral  refractions  which  affect  measurements  of  hori- 
zontal angles  and  azimuth  observations,  in  which  lines  of  sight  are  close  to  the  ground,  it  seems 
certain  that  the  effects  of  lateral  refraction  upon  transit  time  observations  in  which  all  lines 
of  sight  are  elevated  high  above  the  horizon  must  be  almost  or  quite  inappreciable.  Tin's  is 
probably  the  case  whenever  proper  precautions  are  taken  to  avoid  local  refraction  within  a  few 
feet  of  the  instrument.  If,  however,  the  temperature  within  the  observatory  is  much  above 
that  outside,  or  if  active  chimneys  or  other  powerful  sources  of  heat  are  near  the  observatory, 
warm  columns  of  air  rising  from  or  passing  over  the  observatory  may  produce  a  sensible  lateral 
refraction.  The  lateral  refraction  is  included,  with  many  other  errors  from  wliich  it  can  not 
be  separated,  in  the  culmination  error,  (s,),  estimated  on  pages  38-39. 

In  addition  to  the  lateral  refraction  referred  to  in  the  preceding  paragraph  and  tacitly 
assumed  to  be  constant  during  the  interval  of  a  few  seconds  in  wliich  a  star  is  being  observed 
upon,  there  are  usually  momentary  lateral  refractions  which  serve  merely  to  make  the  apparent 
rate  of  progress  of  the  star  variable  and  to  make  the  observer's  errors  greater  than  they  other- 
wise would  be. 

Among  the  instrumental  errors  in  transit  observations  for  time  may  be  mentioned  those 
arising  from  the  chronograph  and  the  reading  of  the  chronograph  sheet,  from  poor  focusing, 
from  nonverticality  of  the  micrometer  wire  or  of  the  lines  of  the  diaphragm,  from  changes  in 
azimuth  and  colhmation,  from  errors  in  the  measured  collimation,  from  errors  in  the  measured 
inclination,  from  irregularity  of  pivots,  and  from  changes  in  the  rate  of  the  chronometer. 

All  of  these  except  the  first  two  are  included  in  the  culmination  error,  (s^,  as  estimated 
on  pages  38  and  39. 

As  already  noted  the  chronographs  of  the  form  now  used  operate  so  well  that  no  appreci- 
able error  is  introduced  by  the  assumption  that  the  speed  of  the  chronograph  is  constant  between 
successive  breaks  of  the  chronometer.  The  chronograph  sheet  is  read  to  hundredths  of  seconds 
for  the  exchange  of  arbitrary  signals  between  stations  in  telegraphic  longitude  work.  In 
observations  made  with  an  observing  key,  marking  the  times  of  transit  across  the  lines  of  a 
diaphragm,  the  chronograph  record  of  the  observations  is  read  for  each  line  to  the  nearest  0.805. 

'  By  imperfect  construction  is  here  meant  the  failure  to  satisfy  fully  the  rigid  geometric  conditions  imposed  by  theory,  but  necessarily  attained 
out  imperfectly  by  the  instrument  maker,  as,  for  example,  the  condition  that  the  cross  section  of  a  pivot  should  be  a  perfect  circle  and  remain  so. 
Imperfect  construction  is  therefore  not  meant  to  imply  poor  construction,  that  is,  construction  much  below  the  attainable  degree  of  excellence. 


DETERMINATION   OF   TIME.  49 

By  so  doing,  a  probable  error  of  about  ±  0.S01  on  each  single  line  is  introduced  into  the  readings; 
but  this  is  too  small  in  comparison  with  the  other  errors  concerned  in  transit  work  to  warrant 
a  closer  reading.  In  observations  made  with  a  transit  equipped  with  a  transit  micrometer, 
where  20  observations  on  each  star  are  recorded,  the  chronograph  record  of  these  observations 
is  read  to  the  nearest  0.81.  The  probable  error  of  a  single  record  (position  of  micrometer  wire) 
from  this  source  is  about  ±0.S02,  but  the  number  of  such  records  obtained  on  a  star  makes  the 
probable  error  of  the  mean  of  these  observations  less  than  ±0.801,  showing  that  a  closer  reading 
of  the  chronograph  sheet  is  not  justifiable. 

Poor  focusing  of  either  the  objective  or  the  eyepiece  leads  to  increased  accidental  errors 
because  of  poor  definition.  But  poor  focusing  of  the  objective  is  especially  objectionable, 
because  it  puts  the  diaphragm  (or  plane  of  the  micrometer  wire)  and  the  star  image  in  different 
planes,  and  so  produces  parallax.  The  parallax  errors  may  be  avoided  to  a  large  extent  by  keep- 
ing the  eyepiece  centered  carefully  over  the  part  of  the  diaphragm  wliich  is  being  observed 
upon,  if  proper  longitudinal  motion  of  the  eyepiece  is  provided  for  that  purpose. 

If  the  lines  of  the  diaphragm  do  not  make  an  angle  of  exactly  90°  with  the  horizontal  axis 
of  the  telescope  a  star  observed  above  or  below  the  middle  of  the  diaphragm  will  be  observed 
too  late  or  too  early.  A  similar  error  will  be  caused  in  the  case  of  the  transit  micrometer  if  the 
movable  wire  does  not,  in  each  of  its  positions,  make  an  angle  of  90°  with  the  horizontal  axis. 
Errors  from  this  source  may  be  made  very  small  by  careful  adjustment  and  by  observing  within 
the  narrow  limits  given  by  two  horizontal  lines  or  wires. 

The  mean  errors  of  azimuth  and  of  collimation,  being  determined  by  the  time  observations 
themselves,  are  canceled  out  from  the  final  result  with  a  thoroughness  which  depends  upon  the 
success  attained  in  selecting  stars.  The  process  of  elimination  depends  upon  the  assumption 
that  the  error  of  azimuth  remains  constant  during  each  half  set  and  that  the  collimation  error 
remains  constant  during  the  whole  set.  The  changes  in  these  errors  during  the  intervals  named, 
arising  from  changes  of  temperature,  shocks  to  the  instrument,  or  other  causes,  produce  errors 
in  the  final  result.  These  errors  will  evidently  be  smaller  the  more  rapidly  the  observations  are 
made,  the  more  carefully  the  instrument  is  handled,  and  the  more  symmetrical  and  constant 
are  the  temperature  conditions.  In  general,  these  errors  are  small  but  not  inappreciable.  In 
this  connection  the  stability  of  the  pier  on  which  the  instrument  rests  is  of  especial  importance, 
and  also  the  degree  to  which  it  is  protected  from  shocks  such  as,  for  instance,  the  observer's  walk- 
ing in  its  immediate  vicinity,  if  there  is  no  floor  to  the  observatory  or  tent. 

It  is  mainly  in  the  light  of  the  preceding  paragraph  that  the  number  of  stars  to  be  observed 
in  a  time  set  must  be  determined.  If  the  number  of  stars  hi  a  tune  set  and  the  length  of  tune 
over  which  it  extends  be  increased,  the  errors  due  to  accumulated  changes  in  the  azimuth  and 
collimation  are  increased.  On  the  other  hand,  if  the  number  of  stars  is  decreased  below  the 
present  standard  (12)  the  number  of  observations  rapidly  approaches  equality  with  the  number 
of  unknowns  (4),  and  the  accuracy  with  which  the  unknowns  are  determined  decreases  very 
rapidly.  From  these  considerations  it  would  seem  that  12  stars  per  set  is  about  the  most 
advantageous  number  when  the  highest  degree  of  accuracy  is  desired.1  Under  normal  condi- 
tions this  number  involves  the  necessity  of  depending  upon  the  constancy  of  the  instrument  in 
azimuth  for  about  30  minutes  and  in  collimation  for  about  1  hour.  If  greater  accuracy  is 
desired  than  can  be  obtained  from  a  set  of  12  stars,  it  is  necessary  to  continue  observing  half 
sets  of  6  stars  each,  with  a  reversal  of  the  instrument  in  its  wyes  between  each  two  half  sets,  but 
the  number  of  stars  in  a  half  set  should  not  be  materially  increased. 

To  a  considerable  extent  the  preceding  two  paragraphs  also  apply  to  the  inclination  error. 
The  changes  in  inclination  during  each  half  set  produce  errors  in  addition  to  those  arising  from 
uncertainty  as  to  the  mean  inclination,  hence  again  the  desirability  of  rapid  manipulation. 
The  mean  inclination  is  determined  from  the  indications  of  the  striding  level,  which  are  more 
or  less  in  error.  Different  observers  seem  to  differ  radically  as  to  the  probable  magnitude  of 

*  When  only  a  minor  degree  of  accuracy  is  desired,  the  number  of  stars  may,  of  course,  be  much  less  than  12. 
8136°— 13 4 


50  U.   S.    COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.   14. 

errors  from  this  source,  but  the  best  observers  are,  prone  to  use  the  striding  level  with  peat  care. 
However  small  this  error  may  be  under  the  best  conditions  and  most  skillful  manipulations, 
there  can  be  no  doubt  that  careless  handling  of  the  striding  level,  or  a  little  heedlessness  about 
bringing  a  warm  reading  lamp  too  near  it,1  may  easily  make  this  error  one  of  the  largest  affecting 
the  result.  An  error  of  0.0002  inch  in  the  determination  of  the  difference  of  elevation  of  the 
two  pivots  of  a  transit  like  that  shown  in  illustration  No.  1  produces  an  error  of  more  than  0s.  1 
in  the  deduced  time  of  transit  of  a  star  near  the  zenith. 

The  method  of  treating  the  level  readings  given  on  page  22  is  based  upon  two  assumptions: 
First,  that  the  indications  of  the  striding  level  are  not  sufficiently  accurate  to  determine  the 
small  changes  of  inclination  during  the  progress  of  a  half  set,  and,  second,  that  if  (as  is  generally 
the  case)  there  is  any  systematic  difference  between  the  inclination  as  defined  by  level  readings 
with  objective  northward  and  with  objective  southward  the  mean  of  these  two  inclinations  is 
the  required  most  probable  value  corresponding  to  intermediate  positions  of  the  telescope  in 
which  it  points  to  stars  near  the  zenith  (time  stars).  There  may  be  individual  cases  in  which 
the  first  of  these  assumptions  should  be  reversed  and  each  star  transit  reduced  by  using  the  level 
reading  which  is  nearest  to  it  in  time,  upon  the  supposition  that  the  actual  changes  of  incli- 
nation are  so  large  that  the  level  indications  furnish  a  real  measure  of  them.  In  general, 
however,'  the  method  of  treating  the  level  readings  shown  on  pages  21-23  is  probably  the  best. 

The  errors  in  the  computed  time  arising  from  inequality  and  irregularity  of  pivots  are  prob- 
ably negligible  for  first-class  instruments  in  good  condition.  Any  small  error  in  the  adopted 
mean  value  of  the  inequality  will  appear  in  the  computation  with  nearly  its  full  value  in  the 
derived  error  of  collimation,  but  will  be  almost  completely  eliminated  from  the  computed 
chronometer  correction.  It  is  only  the  difference  of  the  irregularities  of  the  two  pivots  which 
affect  the  observed  times,  and  it  should  be  noted  that  corresponding  points  on  the  two  pivots 
are  always  under  about  the  same  pressure  at  the  same  time,  and  that  therefore  irregularities 
due  to  wear  tend  to  be  the  same  for  the  two  pivots. 

Changes  in  the  rate  of  the  chronometer  during  the  progress  of  a  set  of  observations  evidently 
produce  errors  in  the  computed  chronometer  correction  at  the  mean  epoch  of  the  set.  Under 
ordinary  circumstances  such  errors  must  be  exceedingly  small.  If,  however,  an  observer  is 
forced  to  use  a  poor  timepiece,  or  if  clouds  interfere  so  as  to  extend  the  time  required  to  make 
a  set  of  observations  over  several  hours,  this  error  may  become  appreciable. 

The  observer's  errors  are  by  far  the  most  serious  of  any  class  of  errors  in  transit  observations 
for  time.  The  observer  is  subject  to  both  accidental  and  constant2  errors  in  his  observations 
of  the  times  of  transit  and  in  his  readings  of  the  striding  level.  The  level  reading  errors  (such 
as  errors  in  estimating  tenths)  are  inappreciable  in  their  effect  upon  the  computed  time,  but 
the  errors  in  observations  of  time  of  transit  enter  into  the  computed  time  with  full  value.  The 
observer's  accidental  errors  are  estimated  under  the  heading  ''Relative  Weights  to  Transits 
Depending  on  the  Star's  Declination"  (pp.  38  and  39).  His  constant  error  in  estimating  the 

1  The  longitudinal  section  of  the  upper  inner  surface  of  a  level  vial  is  made  as  nearly  a  perfect  circle  as  possible.  If  an  observer  will  consider 
how  great  this  radius  of  curvature  is  in  asensitivestridinglevel  he  will  understand  why  very  small  deformations  of  the  level  vial  by  unequal  changes 
of  temperature  have  a  marked  effect  upon  the  position  of  the  bubble.  The  radius  of  curvature  for  a  level  of  which  each  division  is 2mm  long  and 
equivalent  to  1}  seconds  of  arc  is  more  than  300  m  (about  1000  feet). 

*  In  discussing  errors,  and  especially  when  discussing  them  with  reference  to  their  ultimate  effects,  it  is  quite  important  to  keep  clearly  in  mind 
the  distinctions  between  accidental  errors,  constant  errors,  and  systematic  errors.  A  constant  error  is  one  which  has  the  same  effect  upon  all  the 
observations  of  the  series  or  portion  of  a  series  under  consideration.  Accidental  errors  are  not  constant  from  observation  to  observation;  they  are 
as  apt  to  be  minus  as  plus,  and  they  presumably  follow  the  law  of  error  which  is  the  basis  of  the  theory  ofleast  squares.  A  systematic  error  is  one  of 
which  the  algebraic  sign,  and,  to  a  certain  extent,  the  magnitude,  bears  a  fixed  relation  to  some  condition  or  set  of  conditions.  Thus,  for  example, 
the  phase  error  in  observations  of  horizontal  directions  is  systematic  with  respect  to  the  azimuth  of  the  sun  and  of  the  line  of  sight.  The  expression 
"constant  error"  is  often  used  loosely  in  contradistinction  to  "accidental  error,"  in  such  a  way  as  to  include  both  strictly  constant  errors  and  sys- 
tematic errors.  The  effect  of  accidental  errors  upon  the  final  result  may  be  diminished  by  continued  repetition  of  the  observations  and  by  the  least 
square  method  of  computation.  The  effects  of  constant  errors  and  of  systematic  errors  must  be  eliminated  by  other  processes;  for  example,  by 
changing  the  method  or  program  of  observations,  by  special  investigations  or  special  observations  designed  to  evaluate  a  constant  error  or  to 
determine  the  exact  law  of  a  systematic  error.  The  above  discussion  applies  with  full  force,  in  so  far  as  the  observer  is  directly  concerned,  to  errors 
arising  from  imperfect  perception  or  judgment  rather  than  to  blunders  or  mistakes,  such  as  reading  a  level  five  divisions  wrong  or  estimating  a  Urn? 
one  second  wrong.  If  a  mistake  is  so  large  that  it  is  caught  by  the  checks  which  are  used  for  that  purpose  it  is  usually  without  effect  upon  the 
computed  result,  since  it  is  either  corrected  or  the  observation  concerned  is  rejected.  A  mistake  which  is  not  caught  is,  in  its  effect  upon  the  com- 
puted result,  an  accidental  error  and,  if  proper  checks  have  been  used  to  detect  mistakes,  will  lie  within  the  limits  of  magnitude  of  the  accidental 
errors.  A  similar  distinction  between  instrumental  errors  and  instrumental  blunders  may  be  drawn;  for  example,  a  blunder  rather  than  error  is 
caused  by  the  movement  of  an  objective  which  is  loose  in  its  cell. 


DETERMINATION   OF   TIME.  51 

time  of  transit  when  observing  with  a  key,  or  by  the  eye  and  ear  method,  is  known  as  personal 
equation  and  may  amount  to  half  a  second  or  even  a  whole  second  in  an  extreme  case.  In 
observations  with  a  transit  micrometer  this  error  if  it  exists  at  all  is  very  small  and  may  te 
neglected.  The  personal  equation,  and  the  methods  of  measuring  it  and  of  eliminating  it  from 
the  final  results,  will  be  treated  more  fully  in  connection  with  longitude  determinations.  In 
the  same  place  will  be  found  a  discussion  of  the  data  which  indicate  that  the  personal  equation 
in  observations  made  with  a  transit  micrometer  is  so  small  that  it  may  be  neglected  in  longitude 
work. 

To  sum  up,  it  may  be  stated  that  the  accidental  error  in  the  determination  of  a  chronometer 
correction  from  observations  with  a  portable  transit  instrument  upon  twelve  stars  may  be 
reduced  within  limits  indicated  by  a  probable  error  of  from  ±s.01  to  ±MO.  However,  in 
observations  made  without  the  transit  micrometer  the  chronometer  correction  may  be  subject  to 
u  large  constant  error,  the  observer's  absolute  personal  equation,  which  may  be  many  times  as 
great  as  the  probable  (accidental)  error.  If  the  observations  have  been  made  with  the  transit 
micrometer,  there  is  practically  no  personal  equation,  and  the  results  may  be  considered  free 
from  constant  errors  due  to  that  source. 

OTHER  METHODS   OF   DETERMINING  TIME. 

In  the  field  it  is  sometimes  necessary  to  use  other  instruments  as  transits  for  the  determi- 
nation of  time.  A  theodolite,  when  so  used,  is  apt  to  give  results  of  a  higher  degree  of  accuracy 
than  would  be  expected  from  an  instrument  of  its  size,  unless  one  has  in  mind  that  the  princi- 
pal errors  in  transit  time  observations  are  those  due  directly  to  the  observer.  On  the  other 
hand,  zenith  telescopes  of  the  form  in  which  the  telescope  does  not  swing  in  a  plane  passing 
through  the  vertical  axis  of  the  instrument  have  been  found  to  give  disappointing  results  when 
iised  in  the  meridian  for  time,  perhaps  because  of  the  asymmetry  of  the  instrument  and  of  the 
fact  that  there  can  be  no  reversal  of  the  horizontal  axis  in  its  bearings,  but  only  of  the  instrument 
as  a  whole.  The  time  may,  however,  be  thus  determined  with  sufficient  accuracy  for  use  in 
connection  with  determinations  of  latitude  with  the  zenith  telescope.1 

The  determination  of  time  by  the  use  of  the  transit  in  any  position  out  of  the  meridan  has 
been  advocated,  but  has  not  seemed  advisable.  The  additional  difficulty  of  making  the  com- 
putation, over  that  for  a  transit  nearly  in  the  meridian,  and  other  incidental  inconveniences, 
much  more  than  offset  the  fact  that  the  adjustment  for  putting  the  transit  in  the  meridian  is 
then  unnecessary. 

The  use  of  the  transit  in  the  vertical  plane  passing  through  Polaris  at  the  time  of  observa- 
tion has  been  advocated,  and  has  been  used  to  a  considerable  extent  in  Europe  and  in  Canada. 
It  is  not  used  by  this  Survey.  The  advantage  of  this  method  over  the  meridian  method  is 
that  the  stability  of  the  instrument  is  depended  upon  for  only  about  5  minutes  instead  of  30 
minutes  or  more.  This  method  is  open,  though  to  a  less  extent,  to  the  objections  stated  in 
the  preceding  paragraph  against  the  method  of  observing  in  any  position  out  of  the  meridian. 

If  a  mark  nearly  in  the  meridian  has  been  established  and  its  azimuth  determined  the 
chronometer  correction  may  be  determined  at  noon  within  a  half  second  by  observing  the 
transit  of  the  sun  as  follows:  Point  on  the  meridian  mark  just  before  apparent  noon;  observe 
the  transit  of  the  preceding  limb  of  the  sun  across  the  lines  of  the  diaphragm;  reverse  the 
horizontal  axis  of  the  telescope  and  observe  the  transit  of  the  following  limb  across  the  lines  of 
the  diaphragm.  If  the  transit  micrometer  is  used,  the  west  limb  of  the  sun  is  followed  across 
the  center  of  the  field  by  the  micrometer  wire,  and  then  the  telescope  is  reversed  and  the  east 
limb  is  followed  by  the  wire.  The  record  of  observations  on  each  limb  is  recorded  automatically 
on  the  chronograph.  The  striding  level  should  be  read  just  before  the  transit  of  the  preceding 
limb  and  just  after  the  transit  of  the  following  limb.  The  mean  of  all  the  observed  times  is 
the  chronometer  time  of  transit  of  the  sun's  center  across  the  plane  of  the  instrument.  This 

1  For  methods  of  determining  time  witli  a  zenith  telescope  by  using  it  as  an  equal-altitude  instrument,  see  Coast  Survey  Report  for  1869,  Appen- 
dix No.  12,  pp.  226-232. 


52  U.   S.   COAST   AND   GEODETIC   SUBVEY   SPECIAL   PUBLICATION    NO.    14. 

time  corrected  for  azimuth  error,  as  determined  by  the  pointing  on  the  meridian  mark,  and  for 
inclination,  is  the  chronometer  time  of  the  sun's  transit  across  the  meridian.  During  the 
observations  the  instrument  should  be  sheltered  from  the  direct  rays  of  the  sun.  This  may  be 
done  by  hanging  in  front  of  it  a  cloth  with  a  hole  cut  in  it  opposite  the  objective.  This  method 
of  determining  time  may  sometimes  be  found  desirable  in  connection  with  chronometric  determi- 
nations of  longitude  in  Alaska  when  continuous  cloudy  weather  prevents  star  observations. 

When  setting  up  a  transit  at  a  new  station  it  is  sometimes  difficult  to  get  a  close  approxi- 
mation to  the  local  time  with  which  to  make  the  first  setting  of  the  transit  in  the  meridian. 
The  following  method  has  been  used  to  furnish  a  rough  value  of  the  local  time,  and  makes  it 
possible  to  put  the  instrument  so  closely  in  the  meridian  on  the  initial  trial  that  there  is  almost 
no  time  lost  from  the  regular  observations.  At  a  Little  before  local  noon  commence  observing 
the  sun,  following  it  by  moving  the  telescope  both  in  azimuth  and  altitude.  While  the  sun  is 
still  rising  appreciably,  clamp  the  telescope  in  altitude,  and  mark  the  time  of  the  transit  of  the 
sun's  limbs  across  the  horizontal  wire  of  the  telescope;  then  keeping  the  telescope  fixed  in 
altitude  swing  it  slightly  in  azimuth  to  meet  the  descending  sun  and  mark  the  transit  of  the  sun's 
limbs  across  the  same  wire  as  before.  The  mean  of  the  times  will  be  approximately  the  chronom- 
eter time  of  the  sun's  passage  across  the  local  meridian,  and  the  chronometer  correction  on 
apparent  solar  time  can  be  determined,  and  finally  its  correction  on  local  sidereal  time.  With 
this  correction,  using  an  azimuth  star  first  in  the  final  placing  of  the  instrument  in  azimuth, 
it  will  be  found  that  two  approximations  will  usually  be  all  that  are  required  to  set  the  instrument 
close  enough  for  actual  observations.  With  the  meridian  telescope  form  of  instrument  this 
method  may  be  easily  and  accurately  followed. 

Sextant  observations  for  time  by  measuring  the  altitude  of  the  sun  give  sufficiently  accurate 
results  for  many  purposes.1  For  example,  the  chronometer  correction  may  thus  be  determined 
with  sufficient  accuracy  for  use  in  zenith  telescope  determinations  of  latitude  or  in  observations 
for  azimuth  made  upon  a  circumpolar  star  within  an  hour  of  elongation.  If  a  specially  constructed 
vertical  circle  2  is  used,  illustration  No.  8,  the  time  may  be  determined  from  observed  altitudes 
of  a  star  or  the  sun  with  sufficient  accuracy  for  all  purposes  in  observations  for  latitude  and 
azimuth.  The  sun  or  star  should  be  observed  near  the  prime  vertical  if  possible.  This  is  the 
method  used  at  present  by  nearly  all  the  parties  of  this  Survey  engaged  in  latitude  and  azimuth 
observations.  With  time  obtained  in  this  way  azimuth  observations  may  be  made  on  Polaris 
at  any  hour  angle.  This  method  is  also  used  by  the  field  parties  engaged  in  making  magnetic 
observations.3  As  this  method  is  so  frequently  used  a  sample  record  of  observations  and  of 
the  computations  is  given  below  with  such  explanations  as  are  necessary. 

DESCRIPTION   OF  THE  VERTICAL  CIRCLE  AND   ITS  ADJUSTMENTS. 

The  vertical  circles  in  use  in  the  Coast  and  Geodetic  Survey  are,  in  general  form,  like  that 
shown  in  illustration  No.  8. 

The  instrument  is  practically  a  theodolite  with  the  graduated  circle  in  a  vertical  position 
and  the  axis  horizontal,  with  the  telescope  fastened  rigidly  to  the  alidade.  The  circle  and 
alidade  are  fastened  to  a  horizontal  support  which  rests  upon  the  top  of  a  vertical  axis,  the  latter 
fitting  into  a  stand.  There  is  a  counterpoise  to  the  circle  and  alidade  on  the  opposite  side  of  the 
vertical  axis.  The  stand  has  three  leveling  screws,  and  there  may  be  a  graduated  circle  near  its 
base  for  measuring  horizontal  angles  approximately. 

1  For  convenient  instructions,  formulae,  and  tables  for  sextant  observations  for  time  and  other  approximate  astronomic  methods,  sec  Bowditch's 
American  Practical  Navigator,  published  by  the  U.  S.  Navy  Department. 

'  Such  an  instrument  is  used  in  observing  vertical  angles  or  zenith  distances  in  primary  triangulation.  The  circles  of  these  instruments  are 
from  8  to  10  inches  in  diameter  and  are  graduated  very  accurately. 

1  See  p.  45,  Directions  for  Magnetic  Measurements,  Coast  and  Geodetic  Survey. 


No.  8. 


VERTICAL  CIRCLE. 


DETERMINATION   OF   TIME.  53 

Before  starting  observations  the  usual  adjustments  of  the  eyepiece  and  object  glass  should 
be  made  and  the  crosswires  should  be  brought  approximately  into  the  center  of  the  field.  There 
is  no  adjustment  for  collimation  in  either  the  vertical  or  horizontal  plane.  A  coarse  stride  level 
is  used  to  make  the  horizontal  axis  of  the  circle  truly  horizontal  and,  consequently,  the  circle 
vertical,  and  a  sensitive  level  is  placed  parallel  with  and  fastened  to  the  circle  to  define  a  hori- 
zontal line  through  the  instrument.  If,  after  leveling  by  the  two  levels,  the  instrument  is 
rotated  on  its  vertical  axis  through  180°  and  the  bubbles  remain  on  the  graduated  scales  of  the 
level  vials  then  the  adjustments  for  level  are  satisfactory. 

TIME   FROM  OBSERVATIONS  ON   A   STAR  WITH  A  VERTICAL  CIRCLE. 

When  making  the  observations  the  star's  image  is  brought  into  the  field  of  the  telescope 
and  the  telescope  clamped  with  the  horizontal  wire  slightly  ahead  of  the  star.  As  the  star 
crosses  the  horizontal  wire  the  observer  notes  the  time  of  the  chronometer  by  the  eye-and-ear 
method,  or,  at  the  instant  of  crossing,  he  calls  "Mark"  to  the  recorder,  who  notes  the  chronome- 
ter time.  Readings  are  made  of  the  bubble  of  the  fixed  level  and  of  the  verniers  of  the  vertical 
circle.  The  telescope  is  then  rotated  on  its  horizontal  axis  and  revolved  180°  about  the  vertical 
axis  of  the  instrument.  A  second  observation  is  made  on  the  star  and  the  level  and  vertical 
circle  are  read  again.  These  observations  constitute  one  complete  determination  of  the  time. 
It  is  advisable  to  take  at  least  four  such  sets  of  observations  for  the  determination  of  the  chro- 
nometer correction  if  the  results  are  used  for  primary  azimuth  work  where  Polaris  or  some 
other  close  circumpolar  star  is  observed  at  any  hour  angle. 

If,  upon  revolving  the  instrument  through  180°  in  azimuth  for  the  second  reading  on  the 
star  for  any  one  set,  it  is  found  that  one  end  of  the  bubble  extends  beyond  the  graduations  of 
the  level  vial,  it  may  be  brought  back  by  the  foot  screws  of  the  instrument.  It  should  never  be 
brought  back  to  the  graduations  by  moving  the  tangent  screw  which  controls  the  relation 
between  the  bubble  and  the  graduations  of  the  circle.  In  other  words,  the  relation  between 
the  fixed  level  and  the  vertical  circle  qf  the  instrument  should  remain  undisturbed  during  a  set. 
If  the  level  is  badly  out  of  adjustment,  it  should  be  adjusted  between  sets.  Whenever  practicable 
one.  half  of  the  sets  of  observations  should  be  made  on  a  star  in  the  east  and  the  other  half  on 
a  west  star,  both  stars  being  nearly  in  the  prime  vertical  and  at  about  the  same  elevation,  in 
order  to  eliminate  instrumental  errors  and  errors  due  to  refraction. 

The  above  two  paragraphs  apply  also  to  observations  on  the  sun,  except,  of  course,  the  last 
sentence  of  the  second  paragraph.  The  instrumental  and  refraction  errors  may  be  minimized  by 
observing  the  sun  in  the  morning  and  again  in  the  afternoon  at  about  the  same  angular  distance 
from  the  meridian. 

RECORD   OF   OBSERVATIONS   ON   STARS. 

The  following  record  shows  four  sets  of  observations  with  the  vertical  circle,  all  on  an  eastern 
star.  These  observations  were  made  in  connection  with  primary  azimuth  observations  at  Sears 
triangulation  station  in  Texas.  The  azimuth  observations  and  computations  are  shown  on 
pages  147  to  149  of  this  publication.  It  will  be  noticed  that  the  zenith  distances  of  the  star  cor- 
rected for  level  are  computed  in  the  record. 


54 


Forir.  252. 


U.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.    14. 

Double  zenith  distances* 


IStation:  Sears  triangulation  station.    Observer:  \V.  Bowie.    State:  Texas.    County:  Jones.    Instrument:  Vertical  eircle  No.  46. 

Date:  Dec.  22,  1908.] 


Object 
observed 

Time 

Level 

Circle, 
right 
orleft 

Circle 

read- 
ing 

Vernitrs 

Zenith  dis- 
tance 

Remarks 

O 

E 

A 

// 

40 
50 

00 
00 

30 
20 

20 
20 

B 

ft 
60 
50 

40 
20 

60 

40 

60 
40 

C 

D* 

Mean 

a  Tauri 
a  Tauri 
a  Tauri 
crTaun 

h    m      s 
1    03    49  0 
1    06    02  5 

d 

14  1 
14  4 

d 
12  0 
11  8 

23.8 

11.8 
14.3 

R 
L 

L 
K 

R 
L 

L 
R 

0 

49    57 
50    01 

49    49 
48    59 

48    36 
48    47 

48    31 
47    34 

20 
60 

in 
30 

10 
50 

40 
20 

40.0 
53.3 

06.7 

56.  7 

33.3 
16.7 

40.0 
26.7 

49    59    46.  6 
-  3.0 
43.6 

49    24    01.  7 
0.0 
01.7 

48    41     55.0 
-  5.0 
50.0 

48    03    03.  4 
-  1.5 
01.9 

Sidereal  chronometer  No.  1769  was 
used.    Temperature,  5°  C.    Ba- 
rometer, 716  ram 

Value  of  one  division  of  level  bub- 
ble=2".58 

1    04    55.8 

1    07    05.0 
1    08    28.5 

2S.5 
-4.7 

14.3 
11  8 

1    07    46.8 

1    10    06.5 
1    12    00.5 

26.1 
0.0 

16.8 
13.4 

26.1 

09.5 
12.9 

1    11    03.5 

1    13    14.5 
1    15    13.0 

30.2 
-7.8 

13.2 
14.1 

22.4 

12.8 
12.2 

1    14    13.  8 

27.3 
-2.3 

25.0 

*  Vertical  circle  No.  46  differs  from  the  usual  type  of  this  instrument  in  use  by  the  Survey  in  the  number  of  verniers  and  in  the  numbering  of 
the  graduations  of  the  circle.  There  are  four  verniers  as  a  rule,  and  the  circle  graduations  are  generally  numbered  continuously,  so  that  the  differ- 
ence of  the  two  circle  readings,  Circle  R  and  Circle  L,  gives  the  double  zenith  distance.  No.  46has  only  three  verniers  and  the  verticalcircle  gradu- 
ations arc  numbered  from  0°  to  180°  both  ways  from  the  zenith. 

In  the  column  of  remarks  is  given  such  information  as  is  necessary  for  the  proper  inter- 
pretation of  the  record  by  the  computer.  In  this  column  should  also  be  given  notes  on  any 
unusual  occurrence,  such  as  the  jarring  of  the  instrument  or  the  adjustment  of  the  instrument 
during  the  period  of  observations. 

The  above  form  is  bound  in  books  of  octavo  size,  which  are  furnished  to  field  parties  upon 
request. 

The  level  correction,  which  is  shown  in  the  column  headed  "Level"  and  is  applied  to  the 
observed  zenith  distance  in  the  next  to  the  last  column,  is  computed  by  the  formula: 


When  the  level  graduations  are  numbered  continuously,  the  formula  is: 


in  which  O  and  E  are  the  readings  of  the  level  when  the  larger  numbers  are  at  the  object  end 
of  the  le*vel  vial,  and  d  is  the  value  in  seconds  of  arc  of  one  division  of  the  vial. 

The  formula  used  in  computing  time  from  observations  with  a  vertical  circle  on  a  star  or 
on  the  sun  is 


sn 


sin  _r  t  = 


cos  $  cos  d 


in  which  t  is  the  hour  angle,  d  the  declination,  £  the  zenith  distance  of  the  object  observed,  and 
<f>  is  the  latitude  of  the  station. 

In  the  following  form  (No.  381a)  the  usual  method  of  computation  is  shown.     This  form 
is  designed  especially  for  the  computation  of  time  from  the  observed  altitudes  of  a  star. 


DETERMINATION   OF   TIME. 

Computation  of  time,  observations  on  a  star  with  vertical  circle. 

Form  381  a. 

(State,  Texas.    Station,  Sears  triangulation  station.    Chronometer,  1769  Sidereal.    Date,  Dec.  22,  1908.    Barometer,  716  rr.m. 

Temperature,  5°  C.] 


55 


Star:  a  Tauri 

Star:  a  Tauri 

Ti    m        s 

0             ,             „ 

h   m        s 

0             ,             „ 

Chron.  reading,                    Zenith  dist. 

1    04    55.  8 

49    59    44 

1    07    46.8 

49    24    02 

Refraction 

+  1     06 

+  1    05 

Corrected  Z.  D.=c 

50    00    50 

49    25    07 

log  cos  <f>,                             $, 

9.  9257458 

32    33    31 

9.  9257458 

32    33    31 

log  cos  S,                              3 

9.  9821234 

16    19    37 

9.  9821234 

16    19    37 

log  cos  ^+log  cos  3—  log  D,  (j>—  i 

9.  9078692 

16    13    54 

9.  9078692 

16    13    54 

log  sin  J  [C+(#-«],            i  IC+tt-a)) 

9.  7375385 

33    07    22 

9.  7340593 

32    49    30 

log  sin  j  [C-W-,5)],            j  [:-«-«! 

9.  4632265 

16    53    28 

9.4557230 

16    35     36 

Sum  two  !.->g  sines=log  X, 

9.  2007650 

9.  1897823 

log  N-log  D=log  sin  2  j  t, 

9.  2928958 

9.  2819131 

log  sin  J  (,                            J  <  (arc) 

9.  6464479 

26    17    54 

9.6409566 

25    56    35 

h    m         S 

h    m        s 

<  (time),                               I  (arc) 

3    30    23.2 

52    35    48 

3    27    32.7 

51    53    10 

Right  ascension  of  star, 

4    30    41.  9 

4    30    41.9 

Sidereal  time, 

1    00    18.7 

1     03    09.2 

Chronometer  reading, 

1    04    55.8 

1    07    46.  8 

Chronometer  correction, 

-04    37.1 

-04    37.6 

The  correction  is  plus  if  the  chronometer  is  slow  and  minus  if  fast. 

Carry  all  angles  to  seconds  only,  all  times  to  tenths  of  seconds,  and  all  logarithms  to  seven  decimal  places. 

In  space  below,  compute  rate  of  chronometer,  etc. 


Mean  Epoch 
h     m 
1     10 
4     58 


Star 

ft  Tauri 
3  Geminor. 


Chronometer  correction 
m         s 
-4     37.7 
-4     36.7 


Clock  rate=0'.263  per  hour  losing. 

Tn  the  above  computation  the  correction  for  refraction  was  obtained  from  the  tables  on 
pages  58-59  of  this  publication. 

The  apparent  declination  and  right  ascension  of  the  star  were  obtained  from  the  American 
Ephemeris  and  Nautical  Almanac  for  1908  (the  year  of  observation). 

TIME  FROM  OBSERVATIONS  ON  THE  SUN  WITH  THE  VERTICAL  CIRCLE. 

When  the  sun  is  the  object  observed  upon  a  slightly  different  program  of  observations  is 
required.  The  telescope  is  pointed  on  the  sun's  upper  limb  (the  horizontal  wire  of  the  telescope 
made  tangent  to  the  disk  of  the  sun)  with  the  circle  right  and  immediately  afterward  with  the 
circle  left.  At  each  pointing  the  time  of  contact,  the  level  reading,  and  the  reading  of  the 
vertical  circle  are  noted.  The  letters  R  and  L  (right  and  left)  are  used  to  designate  the  posi- 
tion of  the  circle  with  reference  to  the  vertical  axis  of  the  instrument.  Two  quarter  sets  similar 
to  the  above  are  then  made  in  quick  succession  on  the  sun's  lower  limb,  and  finally  another 
quarter  set  on  the  upper  limb.  These  are  recorded  on  the  form  shown  below,  on  which  are  also 
computed  the  zenith  distances  of  the  sun's  limbs  corrected  for  level. 


56 


Form  252. 


U.   S.   COAST   AND   GEODETIC    SUEVEY   SPECIAL   PUBLICATION    NO.    14. 

Double  zenith  distances. 


[Station   Tilden.    Observer,  W.  Bowie.    State,  Minnesota.    County,  Poik.    Instrument,  Vertical  circle  No.  63.    Date,  Sept.  6  1906.] 


Object  observed 

Time 

Level 

Circle 
right 
or 
left 

Circle 
reading 

Verniers 

Zenith 
distance 

Remarks 

0 

E 

A 

B 

C 

D 

Mean 

Sun's  upper  limb         0 

h     m     s 
8    47    39.  5 

d 
32.3 

d 
11.0 

R 

Value  of  one  division  of 

49    02 

24 

M 

45 

30 

38.2 

0 

8    48    47.0 

07.6 

29.2 

L 

147    30 

36 

08 

15 

06 

15.8 

49    13    48.  8 

the  level  vial  =4"  .00 

24.7 

18.2 

-6.5 

-6.5 

49     13    42.  3 

Sun's  lower  limb          Q 

8    50    12.5 

32.5     11.2 

R 

0 

8    51     17.5 

07.  8     29.  3 

L 

246    26 

24 

21 

00 

36 

20.2 

49    28    02.2 

Chronometer,  Sidereal 

24.7 

18.1 

-6.6 

No.  102 

-6.6 

49    27    55.6 

Temperature,  27°  C 

Sun's  lower  limb         Q 

8    52    57.0 

31.2 

09.8 

R 

Barometer  not  read 

0 

8    53    45.  2 

08.0 

29.8 

L 

344    46 

15 

05 

30 

00 

12.8 

49    09    56.3 

23.2 

20.0 

-3.2 

-3.2 

49    09    53.1 

Sun's  upper  limb         (3 

8    55    08.2 

31.5 

10.0 

R 

0 

8    55    52.0 

09.3 

31.0 

L 

81    32 

54 

81 

48 

45 

57.0 

48    23    22.  1 

22.2 

21.0 

-1.2 

-1.2 

48    23    20.  9 

The  observations  on  the  upper  limb  are  computed  separately  from  those  on  the  lower  limb 
in  order  that  one  may  make  more  exact  corrections  for  refraction. 

Computation  of  time,  observations  on  sun  with  vertical  circle. 

Form  381. 

[Station,  Tilden.    Date,  Sept.  6,  1906.    Chronometer,  Sidereal  102.    Temperature,  27°  C.    Barometer  (not  read).] 


Sun's  upper  limb 

Sun's  lower  limb 

h    m      s 

.      ,      „ 

h    m      s 

0            ,            „ 

Chron.  reading,                    Zenith  dist. 

8    48    13.  2 

49     13    42 

8    50    45.0 

49    27    56 

Chron.  reading,                    Zenith  dist. 

8    55    30.  1 

48    23    21 

8    53    21.1 

49    09    53 

Mean,                                    Mean 

8    51    51.6 

48    48    32 

8    52    03.0 

49     18    54 

Parallax 

07 

-        07 

Refraction 

+  1    03 

+  1    04 

Semidiameter 

+  15    54 

-1.5    54 

Corrected  Z.  D.=  J 

49    05    22 

49    03    57 

log  cos  ^,                               <j> 

9.  8279861 

47    42     16 

9.8279861 

47     42     16 

log  cos  3,                               S 

9.  9970883 

6    37    38 

9.  9970S83 

6    37    38 

log  cos  <4+log  cos  i)=log  D,  4>—3 

9.  8250744 

41    04    38 

9.  8250744 

41     04    38 

log  sin  J  [c+(^-j)J,             J  (c+W_,j)j 

9.8501157 

45    05    00 

8.  8500254 

45    04     17 

log  sin  J  [c_(0-»)]F             J(C-W-J)] 

8.  8442464 

4    00    22 

8.  8429S19 

3    59    40 

Sum  two  log  sines=log  N, 

8.  6943621 

8.  6930073 

log  N-log  D=log  sin  '  i  t, 

8.  8692897 

8.  8679329 

log  sin  J4,                             It  (arc) 

9.4346438 

15     47     10 

9.4339664 

15     45    39 

h     m       s 

h     m       s 

((time),                              «(arc) 

2    06     17.3 

31    34    20 

2    06    05.2 

31    31     IS 

Local  apparent  time, 

21     53    42.7 

21    53    54.8 

Equation  of  time, 

-1     31.4 

-1    31.4 

Local  mean  time, 

21    52     11.3 

21    52    23.4 

Local  sidereal  time, 

8    51    33.8 

8    51     45.9 

Chronometer  reading, 

8    51    51.6 

8    52    03.0 

Chronometer  correction 

-17.8 

-17.1 

ft        m 

Longitude  from  Greenwich,  =6     25.3 

Estimated  local  mean  time  of  observation,  =9     52 

Greenwich  mean  time  of  observation,  =4     17 

Interpolation  interval,  from  Greenwich  mean  noon,  =4.3  hours 


h       m 
=6     25.3 
=9     53 
=4     18 
=4.3  hours. 


DETERMINATION    OF    TIME.  57 

In  this  computation  the  correction  for  refraction  was  obtained  from  the  tables  on  pages  58-59 
of  this  publication.  The  argument  used  was  the  apparent  altitude. 

The  first  table  gives  the  mean  refraction,  or  the  refraction  under  an  assumed  standard 
condition  of  760  mm.  ( =  29.  9  in.)  pressure  and  10°  C.  ( =  50°  F.)  temperature. 

The  second  table  gives  the  factor  CB,  by  which  the  mean  refraction  as  obtained  from  the 
first  table  must  be  multiplied,  on  account  of  a  barometer  reading  different  from  760  mm. 

In  the  third  table  is  obtained  the  factor  CT  by  which  the  mean  refraction  must  be  multiplied 
on  account  of  a  temperature  different  from  the  standard  (10°  C.). 

The  resulting  refraction  is  then  r  =  ru  X  CB  X  CT  in  which  ru  is  the  refraction  under  standard 
conditions  obtained  from  the  first  table  and  CB  and  CT  are  the  factors  obtained  from  the  second 
and  third  tables,  respectively.1 

The  reduction  for  semidiameter,  and  the  values  for  the  sun's  declination  and  for  the  equa- 
tion of  time  were  obtained  from  the  American  Ephemeris  and  Nautical  Almanac  for  1906  (the 
year  of  observations). 

The  parallax  was  obtained  from  the  table  on  page  60,  which  was  also  taken  from  Hayford's 
Geodetic  Astronomy. 

The  semidiameter  was  obtained  from  page  405  of  the  Ephemeris. 

The  declination  and  the  equation  of  time  were  obtained  from  pages  146  and  147  of  the 
Ephemeris.  The  interpolation  of  these  quantities  for  the  time  of  observation  is  made  by  the 
use  of  the  interpolation  interval  obtained  at  the  bottom  of  the  computation. 

The  mean  of  the  observations  on  either  limb,  reduced  for  parallax,  refraction,  and  semi- 
diameter  gives  the  true  zenith  distance  of  the  sun's  center.  The  computation  is  by  the  same 
formula  as  is  given  for  the  reduction  of  the  observations  on  a  star.  (See  p.  54.) 

As  the  above  observations  were  made  using  a  sidereal  chronometer,  and  as  the  correction 
on  sidereal  time  was  required,  it  was  necessary  to  reduce  the  computed  mean  time  of  the  observa- 
tion to  its  corresponding  local  sidereal  time  before  a  comparison  was  made  with  the  time  as 
read  from  the  chronometer  face.  The  following  computation  shows  the  various  steps  of  this 
reduction  for  the  observations  on  the  sun's  upper  limb: 

h     m      s 

Local  mean  time  of  observation  (Sept.  5,  1906)  2  21    52     11. 3 

Reduction  to  sidereal  interval  (Table  III,  Ephemeris)  3    35.  6 

Right  ascension  of  mean  sun,  Greenwich  mean  noon  September  5,  1906  10    54    43.  6 

Increase  in  right  ascension  of  mean  sun,  at  Tilden  mean  noon  September  5,  1906 
(Table  III,  Ephemeris,  6"  25m.3  west)  1    03. 3 

Sum,  local  sidereal  time  of  observation  at  Tilden  8    51    33.  8 

For  several  reasons  the  observations  on  a  star  are  more  satisfactory  than  those  on  the  sun. 
When  used  in  connection  with  other  astronomic  observations,  such  as  the  determination  of 
azimuth,  a  chronometer  correction  from  observations  on  a  star  may  be  obtained  close  to  the 
epoch  of  the  observations,  since  any  one  of  many  available  stars  may  be  used.  The  computa- 
tion is  more  easily  made  as  there  is  no  reduction  for  semidiameter  or  for  parallax,  and  the 
declination  and  right  ascension  of  a  star  are  practically  constant  during  an  entire  set  of  observa- 
tions and  therefore  easily  and  quickly  obtained  from  a  star  list.  No  equation  of  time  is  intro- 
duced. 

The  observer  should  have  a  star  chart 3  for  use  in  identifying  the  stars  observed  upon. 

1  These  tables  were  copied  from  A  Text  Book  of  Geodetic  Astronomy  by  John  F.  Hayford,  formerly  inspector  of  geodetic  work  and  Chief  of 
the  Computing  Division,  U.  S.  Coast  and  Geodetic  Survey.  John  Wiley  &  Sons,  1898. 

3  It  must  be  remembered  that  the  day  of  the  Ephermis  is  astronomic,  and  begins  at  noon  of  the  civil  day  of  the  same  date.  Sept.  5,  21&  52w» 
11O,  astronomic  mean  time  is  the  forenoon  of  Sept.  S,  civil  time. 

8  Star  Charts  are  published  by  the  Hydrographic  Office  of  the  U.  S.  Navy  and  may  be  obtained  from  the  Navy  Department,  Washington, 
D.  C.  Star  Charts  are  also  contained  in  A  Field  Book  of  the  Stars,  by  W.  T.  Olcott  (G.  P.  Putnam's  Sons,  publishers). 


58 


U.   S.   COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 

Mean  refraction  (rM) 

[Barometer, 760  millimeters  (=29.9  inches).    Temperature,  10°  C.(=50°  F).] 


Alti- 
tude 

Mean  re- 
fraction 

Change 
per 
minute 

Alti- 
tude 

Mean  re- 
fraction 

Change 
per 
minute 

Alti- 
tude 

Mean  re- 
fraction 

Change 
per 
minute 

Alti- 
tude 

Mean  re- 
fraction 

Change 
per 
minute 

Alti- 
tude 

Mean  re- 
fraction 

Change 
minute 

0    00 

34    08.6 

11.66 

7    00 

7    24.2 

0.95 

19    00 

2    47.6 

0.16 

33    00 

1    29.4 

0.06 

52    30 

0    44.  7 

0.03 

10 

32    15.9 

10.88 

10 

7    14.9 

0.91 

20 

2    44.6 

0.15 

20 

1     28.  2 

0.06 

53    00 

0    43.9 

0.03 

20 

30    31.1 

10.10 

20 

7    06.0 

0.88 

40 

2    41.6 

0.15 

40 

1    27.1 

0.05 

30 

0    43.1 

0.03 

30 

23    53.9 

9.64 

30 

6    57.4 

0.84 

20    00 

2    38.7 

0.14 

34    00 

1    26.1 

0.05 

54    00 

0    42.3 

0.03 

40 

27     18.  2 

9.20 

40 

6    49.1 

0  81 

20 

2    35.9 

0.14 

20 

1    25.0 

0.05 

30 

0    41.6 

0.03 

50 

25    49.8 

8.50 

50 

6    41.2 

0.78 

40 

2    33.2 

0.13 

40 

1    24.0 

0.05 

55    00 

0    40.8 

0.03 

1     00 

24    28.3 

7.82 

8    00 

6    33.5 

0.76 

21    00 

2    30.6 

0.13 

35    00 

1    23.0 

0.05 

30 

0    40.0 

0.03 

10 

23    13.5 

7.17 

10 

6    26.0 

0  73 

20 

2    28.1 

0.13 

20 

1    22.0 

0.05 

56    00 

0    39.3 

0.  025 

20 

22    04.9 

6.58 

20 

6    18.9 

0.70 

40 

2    25.6 

0.12 

40 

1    21.0 

0.05 

57    00 

0    37.8 

0.024 

30 

21    01.  8 

6.06 

30 

6    12.0 

0.68 

22    00 

2    23.2 

0.12 

36    00 

1    20.0 

0.05 

58    00 

0    36.4 

0.023 

40 

20    03.7 

5.60 

40 

6    05.3 

0.66 

20 

2    20.9 

0.12 

30 

1    18.5 

0.05 

59    00 

0    35.0 

0.  023 

50 

19    09.8 

5.20 

50 

5    58.9 

0  63 

40 

2    18.6 

0.11 

37    00 

1    17.1 

0.04 

60    00 

0    33.  6 

0.022 

2    00 

18    19.7 

4.84 

9    00 

5    52.7 

0.61 

23    00 

2    16.4 

0.11 

30 

1     15.7 

0  04 

61    00 

0    32.3 

0.022 

10 

17    33.1 

4.50 

20 

5    40.8 

0.58 

20 

2    14.2 

0.11 

38    00 

1     14.4 

0.04 

62    00 

0    31.0 

0.022 

20 

16    49.7 

4.18 

40 

5    29.7 

0.54 

40 

2     12.1 

0.10 

30 

1     13.1 

0.04 

63    00 

0    29.7 

0.022 

30 

16    09.5 

3.88 

10    00 

5    19.2 

0.51 

24    00 

2    10.1 

0.10 

39    00 

1     11.8 

0.04 

64    00 

0    28.4 

0.021 

40 

15    32.  1 

3.62 

20 

5    09.4 

0  48 

20 

2    08.1 

0.10 

30 

1     10.5 

0.04 

65    00 

0    27.2 

0.021 

50 

14    57.1 

3.39 

40 

5    00.1 

0.46 

40 

2    06.  1 

0.10 

40    00 

1    09.3 

0.04 

66    00 

0    25.9 

0.021 

3    00 

14    24.3 

3.18 

11    00 

4    51.2 

0.43 

25    00 

2    04  2 

0.09 

30 

1    08.1 

0.04 

67    00 

0    24.7 

0.020 

10 

13    53.6 

2.98 

20 

4    42.8 

0.40 

20 

2    02.4 

0.09 

41    00 

1    06.9 

0.04 

68    00 

0    23.6 

0.  020 

20 

13    24.8 

2.79 

40 

4    35.0 

0.38 

40 

2    00.6 

0.09 

30 

1    05.  7 

0.04 

69    00 

0    22.4 

0.020 

30 

12    57.8 

2.61 

12    00 

4    27.5 

•    0.37 

26    00 

1    58.8 

0.09 

42    00 

1    04.6 

0.04 

70    00 

0    21.2 

0.019 

40 

12    32.5 

2.46 

20 

4    20.3 

0.35 

20 

1    57.1 

0.09 

30 

1     03.5 

0.04 

71     00 

0    20.1 

0.019 

50 

12    08.7 

2.33 

40 

4    13.5 

0.33 

40 

1    55.4 

0.08 

43    00 

1    02.4 

0.04 

72    00 

0    18.9 

0.019 

4    00 

11    46.0 

2.20 

13    00 

4    07.1 

0.32 

27    00 

1    53.8 

0.08 

30 

1    01.3 

0.04 

73    00 

0     17.8 

0.018 

10 

11    24.6 

2.09 

20 

4    00.  9 

0.30 

20 

1    52.2 

0.08 

44    00 

1     00.2 

0.03 

74    00 

0    16.7 

0.018 

20 

11    04.2 

1.98 

40 

3    55.1 

0.28 

40 

1    50.6 

0.08 

30 

0    59.2 

0.03 

75    00 

0    15.6 

0.018 

30 

10    44.9 

1.88 

14    00 

3    49.5 

0.27 

28    00 

1    49.1 

0  08 

45    00 

0    58.2 

0.03 

76    00 

0    14.5 

0.018 

40 

10    26.  5 

1.79 

20 

3    44.2 

0.26 

20 

1    47.6 

0.07 

30 

0    57.2 

0.03 

77    00 

0    13.5 

0.018 

50 

10    09.1 

1.70 

40 

3    39.1 

0.25 

40 

1    46.1 

0.07 

46    00 

0    56.2 

0.03 

78    00 

0    12.4 

0.018 

5    00 

9    52.6 

1.61 

15    00 

3    34.1 

0.24 

29    00 

1    44.6 

0.07 

30 

0    55.2 

0.03 

79    00 

0     11.3 

0.018 

10 

9    36.9 

1.54 

20 

3    29.4 

0.23 

20 

1    43.2 

0.07 

47    00 

0    54.2 

0.03 

80    00 

0     10.  3 

0.018 

20 

9    21  9 

1.46 

40 

3    24.8 

0.23 

40 

1    41.8 

0.07 

30 

0    53.3 

0.03 

81     00 

0    09.2 

0.018 

30 

9    07.6 

1.40 

16    00 

3    20.4 

0.22 

30    00 

1    40.5 

0.07 

48    00 

0    52.5 

0.03 

82    00 

0    08.2 

0.018 

40 

8    54.0 

1.33 

20 

3    16.1 

0.21 

20 

1    39.1 

0.07 

30 

0    51.6 

0.03 

83    00 

0    07.2 

0.018 

50 

8    41.0 

1.27 

40 

3    12.0 

0.20 

40 

1    37.8 

0.06 

49    00 

0    50.  7 

0.03 

84    00 

0    06.  1 

0  018 

6    00 

8    28.6 

1.22 

17    00 

3    08.2 

0.19 

31    00 

1    36.6 

0.06 

30 

0    49.8 

0.03 

85    00 

0    05.  1 

0.018 

10 

8     16.7 

1.16 

20 

3    04  5 

0.19 

20 

1    35.3 

0.06 

50    00 

0    48.9 

0.03 

86    00 

0    04.  1 

0.017 

20 

8    05.3 

1.12 

40 

3    00.9 

0.18 

40 

1    34.1 

0.06 

30 

0    48.0 

0.03 

87    00 

0    03.1 

0.017 

30 

7    54.3 

1.07 

18    00 

2    57.4 

0.17 

32    00 

1    33.0 

0.06 

51     00 

0    47.2 

0.03 

88    00 

0    02.0 

0.017 

40 

7    43.  9 

1.02 

20 

2    54.  0 

0.17 

20 

1    31.8 

0.06 

30 

0    46.3 

0.03 

89    00 

0    01.  0 

0.017 

50 

7    33.9 

0.98 

40 

2    50.  7 

0.16 

40 

1    30.6 

0.06 

52    00 

0    45.5 

0.03 

90    00 

0    00.0 

0.017 

DETERMINATION    OF    TIME.  59 

Correction  to  mean  refraction  as  given  on  page  58,  depending  upon  the  reading  of  the  'barometer. 


Barometer 

CB 

Barometer 

CB 

Barometer 

c, 

Barometer 

CB 

Barometer 

CB 

Inches 

mm 

Inches 

mm 

Inches 

mm 

Inches 

mm 

Inches 

mm 

20.0 

508 

0.670 

22.4 

569 

0.749 

24.8 

630 

0.829 

27.2 

691 

0.909 

29.6 

752 

0.989 

20.1 

511 

0.673 

22.5 

572 

0.752 

24.9 

632 

0.832 

27.3 

693 

0.912 

29.7 

754 

0.992 

20.2 

513 

0.676 

22.6 

574 

0.755 

25.0 

635 

0.835 

27.4 

696 

0.916 

29.8 

757 

0.996 

20.3 

516 

0.679 

22.7 

576 

0.759 

25.1 

637 

0.838 

27.5 

699 

0.920 

29.9 

759 

0.999 

20.4 

518 

0.6&2 

22.8 

579 

0.762 

25.2 

640 

0.842 

27.6 

701 

0.923 

30.0 

762 

1.003 

20.5 

521 

0.685 

22.9 

582 

0.766 

25.3 

643 

0.846 

27.7 

704 

0.926 

30.1 

765 

1.007 

20.6 

523 

0.688 

23.0 

584 

0.770 

25.4 

645 

0.849 

27.8 

706 

0.929 

30.2 

767 

1.010 

20.7 

526 

0.692 

23.1 

587 

0.773 

25.5 

648 

0.853 

27.9 

709 

0.933 

30.3 

770 

1.013 

20.8 

528 

0.696 

23.2 

589 

0.776 

25.6 

650 

0.856 

28.0 

711 

0.936 

30.4 

772 

1.016 

20.9 

531 

0.699 

23.3 

592 

0.779 

25.7 

653 

0.859 

28.1 

714 

0.939 

30.5 

775 

1.020 

21.0 

533 

0.  703 

23.4 

594 

0.783 

25.8 

655 

0.862 

28.2 

716 

0.942 

30.6 

777 

1.023 

21.1 

536 

0.706 

23.5 

597 

0.786 

25.9 

658 

0.866 

28.  3 

719 

0.946 

30.7 

780 

1.026 

21.2 

538 

0.709 

23.6 

599 

0.789 

26.0 

660 

0.869 

28.4 

721 

0.949 

30.8 

782 

1.029 

21.3 

541 

0.712 

23.7 

602 

0.792 

26.1 

663 

0.872 

28.5 

724 

0.953 

30.9 

785 

1.033 

21.4 

544 

0.716 

23.8 

605 

0.796 

26.2 

665 

0.875 

28.6 

726 

0.956 

31.0 

787 

1.036 

21.5 

546 

0.719 

23.9 

607 

0.799 

26.3 

668 

0.879 

28.7 

729 

0.959 

21.6 

549 

0.722 

24.0 

610 

0.803 

26.4 

671 

0.882 

28.8 

732 

0.963 

21.7 

551 

0.725 

24.1 

612 

0.806 

26.5 

673 

0.885 

28.9 

734 

0.966 

21.8 

554 

0.729 

24.2 

615 

0.809 

26.6 

676 

0.889 

29.0 

737 

0.970 

21.9 

556 

0.732 

24.3 

617 

0.813 

26.7 

678 

0.892 

29.1 

739 

0.973 

22.0 

559 

0.735 

24.4 

620 

0.816 

26.8 

681 

0.896 

29.2 

742 

0.976 

22.1 

561 

0.739 

24.5 

622 

0.820 

26.9 

683 

0.899 

29.3 

744 

0.979 

22.2 

564 

0.742 

24.6 

625 

0.823 

27.0 

686 

0.902 

29.4 

747 

0.983 

22.3 

566 

0.746 

24.7 

627 

0.826 

27.1 

688 

0.905 

29.5 

749 

0.986 

Correction  to  mean  refraction  as  given  on  page  58,  depending  upon  the  reading  of  the  detached 

thermometer. 


Temperature 

CT 

Temperature 

CT 

Temperature 

CT 

Temperature 

CT 

Temperature 

CT 

Fahren- 
heit 

Centi- 
grade 

Fahren- 
heit 

Centi- 
grade 

Fahren- 
heit 

Centi- 
grade 

Fahren- 
heit 

Centi- 
grade 

Fahren- 
heit 

Centi- 
grade 

-25 

-31.7 

1.172 

8 

-13.3 

1.089 

41 

5.0 

1.018 

74 

23.3 

0.955 

107 

41.7 

0.900 

—24 

-31.1 

1.169 

9 

-12.8 

1.087 

42 

5.6 

1.016 

75 

23.9 

0.953 

108 

42.2 

0.899 

-23 

-30.6 

1.166 

10 

-12.2 

1.085 

43 

6.1 

1.014 

76 

24.4 

0.952 

109 

42.8 

0.897 

-22 

—30.0 

1.164 

11 

-11.7 

1.082 

44 

6.7 

1.012 

77 

25.0 

0.950 

110 

43.3 

0.895 

—21 

—29.4 

1.161 

12 

—11.1 

1.08C 

45 

7.2 

1.010 

78 

25.6 

0.948 

Til 

43.9 

0.894 

-20 

-28.9 

1.158 

13 

-10.6 

1.078 

46 

7.8 

1.008 

79 

26.1 

0.946 

112 

44.4 

0.892 

-19 

-28.3 

1.156 

14 

-10.0 

1.076 

47 

8.3 

1.006 

80 

26.7 

0.945 

113 

45.0 

0.891 

-18 

-27.8 

1.153 

15 

-9.4 

1.073 

48 

8.9 

1.004 

81 

27.2 

0.943 

114 

46.6 

0  890 

-17 

-Z7.2 

1.151 

16 

-  8.9 

1.071 

49 

9.4 

1.002 

82 

27.8 

0.941 

115 

46.1 

0.888 

—16 

-26.7 

1.148 

17 

-  8.3 

1.069 

59 

10.0 

1.000 

83 

28.3 

0.939 

116 

46.7 

0.886 

—  15 

-26.1 

1.145 

18 

-  7.8 

1.067 

51 

10.6 

0.998 

84 

28.9 

0.938 

117 

47.2 

0.885 

-14 

-25.6 

1.143 

19 

-  7.2 

1.064 

52 

11.1 

0.996 

85 

29.4 

0.936 

118 

47.8 

0.884 

—13 

—25.0 

1.140 

20 

-6.7 

1.062 

53 

11.7 

0.994 

86 

30.0 

0.934 

119 

48.3 

0.882 

—12 

—24.4 

1.138 

21 

-  6.1 

1.060 

54 

12.2 

0.992 

87 

30.6 

0.933 

120 

48.9 

0.881 

-11 

-23.9 

1.135 

22 

-  5.6 

1.058 

55 

12.8 

0.990 

88 

31.1 

0.931 

121 

49.4 

0.880 

—  10 

—23.3 

1.133 

23 

-  5.0 

1.056 

56 

13.3 

0.988 

89 

31.7 

0.929 

122 

50.0 

0.878 

Q 

-22.8 

1.130 

24 

-  4.4 

1.054 

57 

13.9 

0.986 

90 

32.2 

0.  928 

123 

50.6 

0.877 

-  8 

—22.2 

1.128 

25 

-  3.9 

1.051 

58 

14.4 

0.985 

91 

32.8 

0.926 

124 

51.1 

0.876 

—  7 

-21.7 

1.125 

26 

-3.3 

1.049 

59 

15.0 

0.983 

92 

33.3 

0.924 

125 

51.7 

0.874 

—  6 

-21.1 

1.123 

27 

-  2.8 

1.047 

60 

15.6 

0.981 

93 

33.9 

0.923 

126 

52.2 

0.873 

-  5 

-20.6 

1.120 

28 

-2.2 

1.045 

61 

16.1 

0.979 

94 

34.4 

0.921 

127 

52.8 

0.871 

—  4 

-20.0 

1.118 

29 

—  1.7 

1.043 

62 

16.7 

0.977 

95 

35.0 

0.919 

128 

53.3 

0.870 

—  3 

-19.4 

1.115 

30 

-  1.1 

1.041 

63 

17.2 

0.975 

96 

35.6 

0.917 

129 

53.9 

0.868 

2 

-18.9 

1.113 

31 

-  0.6 

1.039 

64 

17.8 

0.973 

97 

36.1 

0.916 

130 

54.4 

0.867 

_  1 

-18.3 

1.111 

32 

0.0 

1.036 

65 

18.3 

0.972 

98 

36.7 

0.914 

0 

-17.8 

1.108 

33 

+  0.6 

1.034 

66 

18.9 

0.970 

99 

37.2 

0.912 

+  1 

-17.2 

1.106 

34 

1.1 

1.032 

67 

19.4 

0.968 

100 

37.8 

0.911 

2 

-16.7 

1.103 

35 

1.7 

1.030 

68 

20  0 

0.966 

101 

38.3 

0.909 

3 

-16.1 

1.101 

36 

2.2 

1.028 

69 

20.6 

0.964 

102 

38.9 

0.908 

4 

-15.6 

1.099 

37 

2.8 

1.026 

70 

21  1 

0.962 

103 

39.4 

0.906 

5 

-15.0 

1.096 

38 

3.3 

1.021 

71 

21.7 

0.961 

104 

40  0 

0.905 

6 

-14.4 

1.094 

39 

3.9 

1.022 

72 

22  2 

0.959 

105 

40.6 

0.903 

7 

-13.9 

1.092 

40 

4.4 

1.0?0 

73 

22.8 

0.957 

106 

41.1 

0.902 

60 


U.   S.   COAST   AND   GEODETIC   SUSVEY   SPECIAL   PUBLICATION    NO.   14. 

The  parallax  of  the  sun  (p)  for  the  first  day  of  each  month. 


Altitude 

Jan.  1 

Feb.l 
Dec.l 

Mar.  1 
Nov.  1 

Apr.  1 
Oct.l 

May  1 
Sept,'  1 

June  1 
Aug.  1 

July  1 

Zenith 
distance 

0 

9.0 

9.0 

8.9 

8.9 

8.8 

8.7 

8.7 

90 

3 

9.0 

9.0 

8.9 

8.8 

8.8 

8.7 

8.7 

87 

6 

9.0 

8.9 

8.9 

8.8 

8.7 

8.7 

8.7 

84 

9 

8.9 

8.9 

8.8 

8.8 

8.7 

8.6 

8.6 

81 

12 

8.8 

8.8 

8.7 

8.7 

8.6 

8.5 

8.5 

78 

15 

8.7 

8.7 

8.6 

8.6 

8.5 

8.4 

8.4 

75 

18 

8.6 

8.6 

8.5 

8.4 

8.4 

8.3 

8.3 

72 

21 

8.4 

8.4 

8.3 

8.3 

8.2 

8.2 

8.1 

69 

24 

8.2 

8.2 

8.2 

8.1 

8.0 

8.0 

8.0 

66 

27 

8.0 

8.0 

8.0 

7.9 

7.8 

7.8 

7.8 

63 

30 

7.8 

7.8 

7.7 

7.  7 

7.6 

7.6 

7.6 

60 

33 

7.6 

7.5 

7.5 

7.4 

7.4 

7.3 

7.3 

57 

36 

7.3 

7.3 

7.2 

7.2 

7.1 

7.1 

7.0 

54 

39 

7.0 

7.0 

6.9 

6.9 

6.8 

6.8 

6.8 

51 

42 

6.7 

6.7 

6.6 

6.6 

6.5 

6.5 

6.5 

48 

44 

6.5 

6.5 

6.4 

6.4 

6.3 

6.3 

6.3 

46 

46 

6.3 

6.2 

6.2 

6.2 

6.1 

6.1 

6.0 

44 

48 

6.0 

6.0 

6.0 

5.9 

5.9 

5.8 

5.8 

42 

50 

5.8 

5.8 

5.7 

5.7 

5.6 

5.6 

5.6 

40 

52 

5.6 

5.5 

5.5 

5.4 

5.4 

5.4 

5.4- 

38 

54 

5.3 

5.3 

5.2 

5.2 

5.2 

5.1 

5.1 

36 

56 

5.0 

5.0 

5.0 

5.0 

4.9 

4.9 

4.9 
4  6 

34 
32 

58 
60 

4.  8 
4.5 

4.  8 
4.5 

4.5 

4.4 

4.4 

4.4 

4^4 

30 

62 

4.2 

4.2 

4.2 

4.2 

4.1 

4.1 

4.1 

28 

64 

4.0 

3.9 

3.9 

3.9 

3.8 

3.8 

3.8 

26 

66 

3.7 

3.7 

3.6 

3.6 

3.6 

3.6 

3.5 

24 

68 

3.4 

3.4 

3.4 

3.3 

3.3 

3.3 

3.3 

22 

70 

3.1 

3.1 

3.1 

3.0 

3.0 

3.0 

3.0 

20 

72 

2.8 

2.8 

2.8 

2.7 

2.7 

2.7 

2.7 

18 

74 

2.5 

2.5 

2.5 

2.4 

2.4 

2.4 

2.4 

16 

76 

2.2 

2.2 

2.2 

2.1 

2.1 

2.1 

2.1 

14 

78 

1.9 

1.9 

1.9 

1.8 

1.8 

1.8 

1.8 

12 

80 

1.6 

1.6 

1.6 

1.6 

1.5 

1.5 

1.5 

10 

82 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

8 

84 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

0.9 

6 

86 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

4 

88 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

2 

90 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0 

A,  B,  C,  FACTORS. 

These  factors  are  referred  to  in  the  computations  of  time  from  observations  with  the  transit 
on  pages  23  and  25.     Their  arithmetical  values  are  as  follows: 
Azimuth  factor      =  A  =  sin  £  sec  8 
Level  factor  =5  =  cos  £  sec  d 

domination  f  actor  =  C=sec  d 

where  <5  =  declination  and  £  =  zenith  distance  =  <£  -  §  or  c£-  (180°-<?)  for   stars   observed  at 
upper  or  lower  culmination  respectively. 

The  signs  of  the  factors  are  as  follows : 

A  is  plus  except  for  stars  between  the  zenith  and  the  pole. 

B  is  plus  except  for  stars  observed  at  lower  culmination. 

G  is  plus  for  stars  at  upper  culmination  and  minus  for  stars  at  lower  culmination,  when 
observations  are  made  with  the  instrument  in  the  position,  band  (clamp  or  illumination)  west. 

G  is  minus  for  stars  at  upper  culmination  and  plus  for  stars  at  lower  culmination  when  obser- 
vations are  made  with  the  instrument  in  the  position,  band  (clamp  or  illumination)  east. 

These  factors  are  given  to  two  decimal  places  in  the  tables  on  pages  62  to  77,  and  will  be 
found  sufficiently  accurate  whenever  the  errors  of  adjustment,  a,  b,  and  c,  are  not  allowed  to 
exceed  one  second  of  time.  In  1874  this  Survey  published  more  extended  tables,  giving  these 
factors  to  three  decimal  places.  Where,  from  any  cause,  observations  are  made  with  an  instru- 
mental error  abnormally  large  it  is  desirable  to  take  the  corresponding  star  factors  from  the 
more  extended  table  or  to  compute  them. 


DETERMINATION    OF   TIME.  61 

STAR  FACTORS   OBTAINED   GRAPHICALLY. 

For  a  number  of  years  there  has  been  in  use  in  the  Survey  a  nomogram  for  obtaining  graph- 
ically the  star  factors  A,  B,  and  C,  and  also  K,  the  correction  for  diurnal  aberration.  This 
nomogram  was  devised  by  Mr.  C.  R.  Duvall,  a  computer  in  the  Survey.  It  is  not  only  more 
expeditious  than  the  tables,  but  the  elimination  of  the  double  interpolation  which  the  use  of 
the  tables  necessitates  adds  to  the  accuracy  of  the  derived  factor  in  many  cases. 

The  nomogram  is  shown  in  illustration  No.  9,  reduced  in  size.  It  consists  of  two  systems 
of  equidistant  parallel  lines  perpendicular  to  each  other,  a  system  of  arcs  of  equidistant  concen- 
tric circles,  and  a  transparent  arm,  carrying  a  graduated  straight  line  which  revolves  about  the 
common  center  of  the  circles.  The  decimeter  has  been  the  unit  of  length  in  the  nomograms 
used.  The  three  systems  of  lines  are  drawn  at  a  common  distance  apart  of  1  centimeter.  The 
estimated  tenth  of  this  centimeter  space  gives  the  second  decimal  place  in  the  required  factors. 

The  graduated  line  on  the  under  surface  of  the  transparent  arm  passes  through  the  center 
of  the  axis  about  which  the  arm  revolves.  A  secant  graduation  is  made  upon  this  line,  measured 
from  the  center  of  the  axis  of  revolution.  That  is,  the  graduation  corresponding  to  any  angle 
is  at  a  distance  from  the  center  equal  to  the  secant  of  the  angle  in  question.  This  center  of  the 
axis  of  revolution  is  the  common  center  of  the  concentric  circles  and  also  the  origin  of  the  two 
systems  of  parallel  lines. 

The  graduations  on  the  arm  are  for  the  declinations.  In  the  nomograms  used  the  gradua- 
tions have  not  been  carried  beyond  three  decimeters  from  the  center,  which  limits  the  use  of 
the  instrument  to  declinations  from  0°  to  slightly  over  70°. 

The  zenith  distances  are  graduated  on  one  of  the  concentric  circles  at  a  convenient  dis- 
tance from  the  center.  In  the  instrument  shown  in  the  illustration  the  distance  is  25  centime- 
ters. Since  stars  are  never  observed  at  zenith  distances  approaching  90°,  the  upper  part  of 
the  quadrant  is  not  used. 

To  determine  the  factors  A,  B,  and  C  of  a  given  star,  revolve  the  transparent  arm  until 
the  graduated  line  of  the  arm  coincides  with  the  star's  zenith  distance  on  the  graduated  arc. 
Holding  the  arm  in  this  position,  place  a  needle  point  at  that  point  of  the  graduated  line  which 
corresponds  to  the  star's  declination.  The  position  of  this  point  in  the  three  systems  of  equi- 
distant lines  gives  the  three  factors,  A  being  the  ordinate,  B  the  abscissa,  and  C  the  radius 
vector. 

The  nomogram  shown  in  the  illustration  is  of  thin  bristol  board  pasted  smoothly  on  thick 
cardboard.  The  transparent  arm  is  of  celluloid  one-sixteenth  of  an  inch  thick.  The  axis  of 
the  arm  is  a  solid  metal  cylinder  with  a«head  which  fits  against  the  back  of  the  cardboard. 
The  axis  is  made  long  so  that  the  arm  can  be  placed  on  it  and  revolved  without  being  made 
fast. 

The  correction  for  aberration  may  be  taken  from  the  same  nomogram,  as  follows:  Set  the 
revolving  arm  at  that  angle  on  the  graduated  circle  which  is  equal  to  the  latitude  of  the  given 
station.  From  the  graduated  line  of  the  arm  read  off  the  decimation  at  each  intersection  with 
a  broken-line  ordinate.  These  declinations  are  the  limits  between  which  «  has  the  values 
08.00,  08.01,  08.02,  etc.,  for  the  latitude  of  the  station  in  question.  By  means  of  these  limits 
the  K  of  any  star  can  be  immediately  written  down  from  its  declination.  The  broken-line  ordi- 

,  .  .  .005   .015   .025 

nates  are  drawn  at  distances  from  the  origin  equal  to  -TV>T,  "noT*  ~fyrj'         '  '  '  decimeters. 


62 


U.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.    14. 


Table  of  factors  for  reduction  of  transit  observations. 

TOP  ARGUMENT- STAR'S  DECLINATION  (i). 

SIDE   ARGUMENT- STAR'S  ZENITH  DISTANCE  (;). 

[For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  opposite  page.) 


C 

0° 

10° 

15° 

20° 

22° 

24° 

26° 

28° 

30° 

32° 

34° 

36° 

38° 

40° 

41° 

42° 

C 

1 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

89 

2 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.05 

.05 

.05 

88 

3 

.05 

.05 

.05 

.03 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.07 

.07 

.07 

.07 

87 

4 

.07 

.07 

.07 

.07 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.09 

.09 

.09 

.09 

.09 

86 

5 

.09 

.09 

.09 

.09 

.09 

.10 

.10 

.10 

.10 

.10 

.10 

.11 

.11 

.11 

.11 

.12 

85 

6 

.11 

.11 

.11 

.11 

.11 

.11 

.12 

.12 

.12 

.12 

.13 

.13 

.13 

.14 

.14 

.14 

84 

7 

.12 

.12 

.13 

.13 

.13 

.13 

.14 

.14 

.14 

.14 

.15 

.15 

.15 

.16 

.16 

.16 

83 

8 

.14 

.14 

.14 

.15 

.15 

.15 

.16 

.16 

.16 

.16 

.17 

.17 

.18 

.18 

.18 

.19 

82 

9 

.16 

.16 

.16 

.17 

.17 

.17 

.17 

.18 

.18 

.18 

.19 

.19 

.20 

.20 

.21 

.21 

81 

10 

.17 

.18 

.18 

.19 

.19 

.19 

.19 

.20 

.20 

.?! 

.21 

.21 

.22 

.23 

.23 

.23 

80 

11 

.19 

.19 

.20 

.20 

.21 

.21 

.21 

.22 

.22 

.23 

.23 

.24 

.24 

.25 

.25 

.26 

79 

12 

.21 

.21 

.22 

.22 

.22 

.23 

.23 

.24 

.24 

.25 

.25 

.26 

.26 

.27 

.27 

.28 

78 

13 

.22 

.23 

.23 

.24 

.24 

.25 

.25 

.2ti 

.26 

.27 

.27 

.28 

.29 

.29 

.30 

.30 

77 

14 

.24 

.25 

.25 

.28 

.26 

.27 

.27 

.27 

.28 

.29 

.29 

.30 

.31 

.32 

.32 

.33 

76 

15 

.26 

.26 

.27 

.28 

.28 

.28 

.29 

.29 

.30 

.31 

.31 

.32 

.33 

.34 

.34 

.35 

75 

16 

.28 

.28 

.29 

.29 

.30 

.30 

.31 

.31 

.32 

.33 

.33 

.34 

.35 

.36 

.37 

.37 

74 

17 

.29 

.30 

.30 

.31 

.31 

.32 

.33 

.33 

.34 

.34 

.35 

.36 

.37 

.38 

.39 

.39 

73 

18 

.31 

.31 

.32 

.33 

.33 

.33 

.34 

.35 

.36 

.36 

.37 

.38 

.39 

.40 

.41 

.42 

72 

19 

.33 

.33 

.34 

.35 

.35 

.36 

.36 

.37 

.38 

.38 

.39 

.40 

.41 

.42 

.43 

.44 

71 

20 

.34 

.35 

.35 

.36 

.37 

.37 

.38 

.39 

.40 

.40 

.41 

.42 

.43 

.45 

.45 

.46 

70 

21 

.36 

.36 

.37 

.38 

.39 

.39 

.40 

.41 

.41 

.42 

.43 

.44 

.45 

.47 

.47 

.48 

69 

22 

.37 

.38 

.39 

.40 

.40 

.41 

.42 

.42 

.43 

.44 

.45 

.46 

.  .48 

.49 

.50 

.50 

68 

23 

.39 

.40 

.41 

.42 

.42 

.43 

.44 

.44 

.45 

.46 

.47  1    .48 

.50 

.51 

.52 

.53 

67 

24 

.41 

.41 

.42 

.43 

.44 

.45 

.45 

.46 

.47 

.48 

.49 

.50 

.52 

.53 

.54 

.55 

66 

25 

.42 

.43 

.44 

.45 

.46 

.46 

.47 

.48 

.49 

.50 

.51 

.52 

.54 

.55 

.56 

.57 

05 

26 

.44 

.45 

.45 

.47 

.47 

.48 

.49 

.50 

.51 

.52 

.53 

.54 

.56 

.57 

.58 

.59 

64 

27 

.45 

.46 

.47 

.48 

.49 

.50 

.51 

.51 

.52 

.54 

.55 

.56 

.58 

.59 

.CO 

.61 

63 

28 

.47 

.48 

.49 

.50 

.51 

.51 

.52 

.53 

.54 

.55 

.57 

.58 

.CO 

.61 

.62 

.63 

62 

29 

.48 

.49 

.50 

.52 

.52 

.53 

.54 

.55 

.56 

.57 

.58 

.60 

.61 

.63 

.64 

.65 

61 

30 

.50 

.51 

.52 

.53 

.54 

.55 

.56 

.57 

.58 

.59 

.60 

.62 

.63 

.65 

.66 

.67 

60 

31 

.52 

.52 

.53 

.55 

.56 

.56 

.57 

.58 

.59 

.61 

.62 

.64 

.68 

.67 

.68 

.69 

59 

32 

.53 

.54 

.55 

.56 

,57 

.58 

.59 

.60 

.61 

.63 

.64 

.65 

.67 

.69 

.70 

.71 

58 

33 

.55 

.55 

.56 

.58 

.59 

.60 

.61 

.62 

.63 

.64 

.66 

.67 

.69 

.71 

.72 

.73 

57 

34 

.56 

.57 

.58 

.59 

.60 

.61 

.62 

.63 

.65 

.66 

.67 

.69 

.71 

.73 

.74 

.  75 

56 

35 

.57 

.58 

.59 

.61 

.62 

.63 

.64 

.65 

.66 

.68 

.69 

.71 

.73 

.75 

.76 

.77 

55 

36 

.59 

.60 

.61 

.63 

.63 

,64 

.65 

.67 

.68 

.66 

.71 

.73 

.75 

,77 

.78 

.79 

54 

37 

.60 

.61 

.62 

.64 

.65 

.66 

.67 

.68 

.70 

.71 

.73 

.74 

.76 

.79 

.80 

.81 

53 

38 

.62 

.63 

.64 

.66 

.66 

.67 

.69 

.70 

.71 

.73 

.74 

.76 

.78 

.80 

.82 

.83 

52 

39 

.63 

.64 

.65 

.67 

.68 

.69 

.70 

.71 

.73 

.74 

.76 

.78 

.80 

.82 

.83 

.85 

51 

40 

.64 

.65 

.67 

.68 

.69 

.70 

.72 

.73 

.74 

.76 

.77 

.79 

.82 

.84 

.85 

.86 

50 

41 

.66 

.67 

,68 

.70 

.71 

.72 

.73 

.74 

.76 

.77 

.79 

.81 

.S3 

.86 

.87 

.88 

49 

42 

.67 

.68 

.69 

.71 

.72 

.73 

.74 

.76 

.77 

.79 

.81 

.83 

.85 

.87 

.89 

.90 

48 

43 

.68 

.69 

.71 

.73 

.74 

.75 

.76 

.77 

.79 

.80 

.82 

.84 

.86 

.89 

.90 

.92 

47 

44 

.69 

.71 

.72 

.74 

.» 

.76 

.77 

.79 

.80 

.82 

.84 

.86 

.88 

.91 

.92 

.93 

4C 

45 

.71 

.72 

.73 

.75 

.76 

.77 

.79 

.80 

.82 

.83  j   .85 

.87 

.90 

.92 

.94 

.95 

45 

0° 

10° 

15° 

20° 

22° 

24° 

26° 

28° 

30° 

32°  !  34° 

36° 

3S° 

40° 

41° 

42° 

DETERMINATION    OF    TIME. 


63 


Table  of  factors  for  reduction  of  transit  observations. 

TOP  ARGUMENT=STAR'S  DECLINATION  (<»). 

SIDE  ARGUMENT"  STAR'S  ZENITH  DISTANCE  (C). 

[For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  this  page.] 


: 

0° 

10° 

15° 

20" 

22° 

24° 

26° 

28° 

30° 

32° 

34° 

36° 

38° 

40° 

41° 

42° 

C 

0 

0 

4fi 

72 

.73 

.74 

.  77 

.78 

.79 

.80 

.82 

.83 

.85 

.87 

.89 

.91 

.94 

.95 

.97 

44 

47 

.73 

.74 

.76 

.73 

.79 

.80 

.81 

.83 

.84 

.86 

.88 

.90 

.93 

.95 

.97 

.98 

43 

48 

.74 

.76 

.77 

.79 

.80 

.81 

.83 

.84 

.86 

.88 

.90 

.92 

.94 

.97 

.98 

1.00 

42 

49 

.  75 

.77 

.78 

.80 

.81 

.83 

.84 

.86 

.87 

.89 

.91 

.93 

.96 

.99 

1.00 

1.02 

41 

••>» 

.77 

.78 

.79 

.82 

.83 

.84 

.85 

.87 

.89 

.90 

.92 

.95 

.97 

1.00 

1.01 

1.03 

40 

.78 

.79 

.80 

.83 

.84 

.85 

.87 

.88 

.90 

.92 

.94 

.96 

.99 

1.01 

1.03 

1.05 

39 

5 

.79 

.80 

.82 

.84 

.85 

.86 

.88 

.89 

.91 

.93 

.95 

.97 

1.00 

1.03 

1.04 

1.06 

38 

53 

.80 

.81 

.83 

.85 

.86 

.87 

.89 

.91 

.92 

.94 

.96 

.99 

1.01 

1.04 

1.06 

1.07 

37 

54 

.81 

.82 

.84 

.86 

.87 

.89 

.90 

.92 

.93 

.96 

.98 

1.00 

1.03 

1.06 

1.07 

1.09 

36 

55 

.82 

.83 

.85 

.87 

.88 

.90 

.91 

.93 

.95 

.97 

.99 

1.01 

1.04 

1.07 

1.08 

1.10 

35 

56 

.83 

.84 

.86 

.88 

.89 

.91 

.92 

.94 

.96 

.98 

1.00 

1.02 

1.05 

1.08 

1.10 

1.12 

34 

57 

.84 

.85 

.87 

.89 

.90 

.92 

.93 

.95 

.97 

.99 

1.01 

1.04 

1.06 

1.09 

1.11 

1.13 

33 

58 

.85 

.86 

.88 

.90 

.91 

.93 

.94 

.96 

.98 

1.00 

1.02 

1.05 

1.08 

1.11 

1.12 

1.14 

32 

59 

.86 

.87 

.89 

.91 

.92 

.94 

.95 

.97 

.99 

1.01 

1.03 

1.06 

1.09 

1.12 

1.14 

1.15 

31 

60 

.87 

.88 

.90 

.92 

.93 

.95 

.96 

.98 

1.00 

1.02 

1.04 

1.07 

1.10 

1.13 

1.15 

1.17 

SO 

61 

.87 

.89 

.91 

.93 

.94 

.96 

.97 

.99 

1.01 

1.03 

1.05 

1.08 

1.11 

1.14 

1.16 

1.18 

29 

62 

.88 

.90 

.91 

.94 

.95 

.97 

.98 

1.00 

1.02 

1.04 

1.06 

1.09 

1.12 

1.15 

1.17 

1.19 

28 

63 

.89 

.91 

.92 

.95 

.96 

.98 

.99 

1.01 

1.03 

1.05 

1.07 

1.10 

1.13 

1.16 

1.18 

1.20 

27 

64 

.90 

.91 

.93 

.96 

.97 

.98 

1.00 

1.02 

1.04 

1.06 

1.08 

1.11 

1.14 

1.17 

1.19 

1.21 

26 

65 

.91 

.92 

.94 

.96 

.98 

.99 

1.01 

1.03 

1  05 

1.07 

1.09 

1.12 

1.15 

1.18 

1.20 

1.22 

25 

66 

.91 

.93 

.95 

.97 

.99 

1.00 

1.02 

1.04 

1.06 

1.08 

1.10 

1.13 

1.16 

1.19 

1.21 

1.23 

24 

67 

.92 

.94 

.95 

.98 

.99 

1.01 

1.02 

1.04 

1.06 

1.09 

1.11 

1.14 

1.17 

1.20 

1.22 

1.24 

23 

68 

.93 

.94 

.96 

.99 

1.00 

1.02 

1.03 

1.05 

1.07 

1.09 

1.12 

1.15 

1.18 

1.21 

1.23 

1.25 

22 

69 

.93 

.95 

.97 

.99 

1.01 

1.02 

1.04 

1.06 

1.08 

1.10 

1.13 

1.15 

1.18 

1.22 

1.24 

1.26 

21 

70 

.94 

.95 

.97 

1.00 

1.01 

1.03 

1.05 

1.06 

1.09 

1.11 

1.13 

1.16 

1.19 

1.23 

1.25 

1.26 

•20 

71 

.95 

.96 

.93 

1.01 

1.02 

1.04 

1.05 

1.07 

1.09 

1.12 

1.14 

1.17 

1.20 

1.23 

1.25 

1.27 

19 

72 

.95 

.97 

.98 

1.01 

1.03 

1.04 

1.06 

1.08 

1.10 

1.12 

1.15 

1.17 

1.21 

1.24 

1.26 

1.28 

18 

73 

.96 

.97 

.99 

1.02 

1.03 

1.05 

1.06 

1.08 

1.10 

1.13 

1.15 

1.18 

1.21 

1.25 

1.27 

1.29 

17 

74 

.96 

.98 

1.00 

1.02 

1.04 

1.05 

1.07 

1.09 

1.11 

1.13 

1.16 

1.19 

1.22 

1.25 

1.27 

1.29 

16 

75 

.97 

.98 

1.00 

1.03 

1.04 

1.06 

1.08 

1.09 

1.12 

1.14 

1.16 

1.19 

1.23 

1.26 

1.28 

1.30 

15 

76 

.97 

.99 

1.00 

1.03 

1.05 

1.06 

1.08 

1.10 

.12 

1.14 

1.17 

1.20 

1.23 

1.27 

1.29 

1.31 

14 

77 

.97 

.99 

1.01 

1.04 

1.05 

1.07 

1.08 

1.10 

.13 

1.15 

1.17 

1.20 

1.24 

1.27 

1.29 

1.31 

13 

78 

.98 

.99 

1.01 

1.04 

1.05 

1.07 

1.09 

1.11 

.13 

1.15 

1.18 

1.21 

1.24 

1.28 

1.30 

1.32 

12 

79 

.98 

1.00 

1.02 

1.04 

1.06 

1.08 

1.09 

1.11 

.13 

1.16 

1.18 

1.21 

1.25 

1.28 

1.30 

1.32 

11 

80 

.98 

1.00 

1.02 

1.05 

1.06 

1.08 

1.10 

1.12 

.14 

1.16 

1.19 

1  22 

1.25 

1.29 

1.30 

1.33 

10 

81 

.99 

1  00 

.02 

1.05 

1.07 

1.08 

1.10 

1.12 

.14 

1.17 

1.19 

1.22 

1.25 

1.29 

1.31 

1.33 

9 

82 

.99 

1.01 

.03 

1.05 

1.07 

1.08 

1.10 

1.12 

.14 

1.17 

1.19 

1.22 

1.26 

1.29 

1.31 

1.33 

8 

83 

.99 

1.01 

.03 

1.06 

1.07 

1.09 

1.10 

1.12 

.15 

1.17 

1.20 

1.23 

1.26 

1.30 

1.32 

1.34 

7 

84 

.99 

1.01 

.03 

1.06 

1.07 

1.09 

1.11 

1.13 

.15 

1.17 

1.20 

1.23 

1.26 

1.30 

1.32 

1.34 

6 

85 

1.00 

1.01 

.03 

1  06 

1.07 

1.09 

1.11 

1.13 

.15 

1.17 

1.20 

1.23 

1.26 

1.30 

1.32 

1.34 

5 

86 

1.00 

1.01 

.03 

1.06 

1.08 

1.09 

1.11 

1.13 

.15 

1.18 

1.20 

1.23 

1.27 

1.30 

1.32 

1.34 

4 

87 

1.00 

1.01 

.03 

1.06 

1.08 

1.09 

1.11 

1.13 

.15 

1.18 

1.20 

1.23 

1.27 

1.30 

1.32 

1.34 

3 

88 

1.00 

1.01 

.03 

1.06 

1.08 

1.09 

1.11 

1.13 

.15 

1.18 

1.20 

1.23 

1.27 

1.30 

1.32 

1.34 

2 

89 

1.00 

1.02 

.04 

1.06 

1.03 

1.09 

1.11 

1.13 

.15 

1.18 

1.21 

1.24 

1.27 

1.31 

1.32 

1.35 

1 

90 

1.00 

1.02 

.04 

1.06 

l.OS 

1.09 

1.11 

1.13 

.15 

1.1S 

1.21 

1.24 

1.27 

1.31 

1.32 

1.35 

0 

0° 

10° 

15° 

20° 

22° 

24° 

26° 

28° 

30° 

32° 

34° 

36° 

38° 

40° 

41° 

42° 

64  U.    S.    COAST    AND   GEODETIC    SUEVEY    SPECIAL    PUBLICATION    NO.    14. 

Table  of  factors  for  reduction  of  transit  observations. 

TOP  ARGUMENT- STAR'S  DECLINATION  (<»). 

SIDE  ARGUMENT- STAR'S  ZENITH  DISTANCE  (C). 

[For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  opposi/e  page.] 


C 

42' 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

50° 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

C 

a 

1 

.02 

.02 

.02 

.02 

.02 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

.03 

89 

2 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

88 

3 

.07 

.07 

.07 

.07 

.07 

.08 

.08 

.08 

.08 

.08 

.08 

.09 

.09 

.09 

.09 

.10 

87 

4 

.09 

.10 

.10 

.10 

.10 

.10 

.10 

.11 

.11 

.11 

.11 

.12 

.12 

.12 

.12 

.13 

86 

5 

.12 

.12 

.12 

.12 

.13 

.13 

.13 

.13 

.13 

.14 

.14 

.14 

.15 

.15 

.16 

.16 

85 

6 

.14 

.14 

.15 

.15 

.15 

.15 

.16 

.16 

.16 

.17 

.17 

.17 

.18 

.18 

.19 

.19 

84 

7 

.16 

.17 

.17 

.17 

.18 

.18 

.18 

.19 

.19 

.19 

.20 

.20 

.21 

.21 

.22 

.22 

83 

8 

.19 

.19 

.19 

.20 

.20 

.20 

.21 

.21 

.22 

.22 

.23 

.23 

.24 

.24 

.25 

.26 

82 

9 

.21 

.21 

.22 

.22 

.22 

.23 

.23 

.24 

.24 

.25 

.25 

.26 

.27 

.27 

.28 

.29 

81 

10 

.23 

.24 

.24 

.25 

.25 

.25 

.26 

.26 

.27 

.28 

.28 

.29 

.30 

.30 

.31 

.32 

80 

11 

.26 

.26 

.27 

.27 

.28 

.28 

.28 

.29 

.30 

.30 

.31 

.32 

.32 

.33 

.34 

.35 

79 

12 

.28 

.28 

.29 

.29 

.30 

.30 

.31 

.32 

.32 

.33 

.34 

.35 

.35 

.36 

.37 

.38 

78 

13 

.30 

.31 

.31 

.32 

.32 

.33 

.34 

.34 

.35 

.36 

.36 

.37 

.38 

.39 

.40 

.41 

77 

14 

.33 

.33 

.34 

.34 

.35 

.35 

.36 

.37 

.38 

.38 

.39 

.40 

.41 

.42 

.43 

.44 

76 

15 

.35 

.35 

.36 

.37 

.37 

.38 

.39 

.39 

.40 

.41 

.42 

.43 

.44 

.45 

.46 

.48 

75 

16 

.37 

.38 

.38 

.39 

.40 

.40 

.41 

.42 

.43 

.44 

.45 

.46 

.47 

.48 

.49 

.51 

74 

17 

.39 

.40 

.41 

.41 

.42 

.43 

.44 

.45 

.45 

.46 

.47 

.49 

.50 

.51 

.52 

.54 

73 

18 

.42 

.42 

.43 

.44 

.44 

.45 

.46 

.47 

.48 

.49 

.50 

.51 

.53 

.54 

.55 

.57 

72 

19 

.44 

.45 

.45 

.46 

.47 

.48 

.49 

.50 

.51 

.52 

.53 

.54 

.55 

.57 

.58 

.60 

71 

20 

.46 

.47 

.48 

.48 

.49 

.50 

.51 

.52 

.53 

.54 

.56 

.57 

.58 

.60 

.61 

.63 

70 

21 

.48 

.49 

.50 

.51 

.52 

.52 

.54 

.55 

.56 

.57 

.58 

.59 

.61 

.62 

.64 

.66 

69 

22 

.50 

.51 

.52 

.53 

.54 

.55 

.56 

.57 

.58 

.60 

.61 

.62 

.64 

.65 

.67 

.69 

68 

23 

.53 

.53 

.54 

.55 

.56 

.57 

.58 

.60 

.61 

.62 

.63 

.65 

.66 

.68 

.70 

.72 

67 

24 

.55 

.56 

.57 

.58 

.59 

.60 

.61 

.62 

.63 

.65 

.66 

.68 

.69 

.71 

.73 

.75 

66 

25 

.57 

.58 

.59 

.60 

.61 

.62 

.63 

.64 

.66 

.67 

.69 

.70 

.72 

.74 

.76 

.78 

65 

26 

.59 

.60 

.61 

.62 

.63 

.64 

.65 

.67 

.68 

.70 

.71 

.73 

.75 

.76 

.78 

.80 

64 

27 

.61 

.62 

.63 

.64 

.65 

.67 

.68 

.69 

.71 

.72 

.74 

.75 

.77 

.79 

.81 

.83 

63 

28 

.63 

.64 

.65 

.66 

.68 

.69 

.70 

.72 

.73 

.75 

.76 

.78 

.80 

.82 

.84 

.86 

62 

29 

.65 

.66 

.67 

.69 

.70 

.71 

.72 

.74 

.75 

.77 

.79 

.81 

.82 

.84 

.87 

.89 

61 

30 

.67 

.68 

.69 

.71 

.72 

.73 

.75 

.76 

.78 

.79 

.81 

.83 

.85 

.87 

.89 

.92 

60 

31 

.69 

.70 

.72 

.73 

.74 

.75 

.77 

.78 

.80 

.82 

.84 

.86 

.88 

.90 

.92 

.95 

59 

32 

.71 

.72 

.74 

75 

.76 

.78 

.79 

.81 

.82 

.84 

.86 

.88 

.90 

.92 

.95 

.97 

58 

33 

.73 

.74 

.76 

.77 

.78 

.80 

.81 

.83 

.85 

.87 

.88 

.91 

.93 

.95 

.97 

1.00 

57 

34 

.75 

.76 

.78 

.79 

.80 

.82 

.84 

.85 

.87 

.89 

.91 

.93 

.95 

.97 

1.00 

1.03 

56 

35 

.77 

.78 

.80 

.81 

.83 

.84 

.86 

.87 

.89 

.91 

.93 

.95 

.98 

1.00 

1.03 

1.05 

55 

36 

.79 

.80 

.82 

.83 

.85 

.86 

.88 

.90 

.91 

.93 

.95 

.98 

1.00 

1.03 

1.05 

1.08 

54 

37 

.81 

.82 

.84 

.85 

.87 

.88 

.90 

.92 

.94 

.96 

.98 

.00 

1.02 

1.05 

1.08 

.10 

53 

38 

.83 

.84 

.86 

.87 

.89 

.90 

.92 

.94 

.96 

.98 

1.00 

.02 

1.05 

1.07 

1.10 

.13 

52 

39 

.85 

.86 

.87 

.89 

.91 

.92 

.94 

.96 

.98 

1.00 

1.02 

.05 

1.07 

1.10 

1.12 

.15 

51 

40 

.86 

.88 

.89 

.91 

.93 

.94 

.96 

.98 

1.00 

1.02 

1.04 

.07 

1.09 

1.12 

1.15 

.18 

50 

41 

.88 

.90 

.91 

.93 

.94 

.96 

.98 

1.00 

1.02 

1.04 

1.07 

.09 

1.12 

1.14 

1.17 

.20 

49 

42 

.90 

.91 

.93 

.95 

.96 

.98 

1.00 

1.02 

1.04 

1.06 

1.09 

.11 

1.14 

1.17 

1.20 

.23 

48 

43 

.92 

.93 

.95 

.96 

.98 

1.00 

1.02 

1.04 

1.06 

1.08 

1.11 

.13 

1.16 

1.19 

1.22 

.25 

47 

44 

.93 

.95 

.97 

.98 

1.00 

1.02 

1.04 

1.06 

1.08 

1.10 

1.13 

1.15 

1.18 

1.21 

1.24 

.28 

46 

45 

.95 

.97 

.98 

1.00 

1.02 

1.04 

1.06 

1.08 

1.10 

1.12 

1.15 

1.17 

1.20 

1.23 

1.26 

.30 

4.5 

42° 

43° 

44° 

45° 

46° 

47° 

4S° 

49° 

50° 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

DETERMINATION   OF   TIME. 

Table  of  factors  for  reduction  of  transit  observations. 

TOP  ARGUMENT- STAR'S  DECLINATION  (<>). 
SIDE  ARGUMENT- STAR'S  ZENITH  DISTANCE   (C). 
factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  this  page.l 


65 


C 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

50° 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

C 

46 

.97 

.98 

1.00 

1.02 

1.04 

1.05 

1.07 

1.10 

1.12 

1.14 

1.17 

1.19 

1.22 

1.25 

1.29 

1.32 

44 

47 

.98 

1.00 

1.02 

1.03 

1.05 

1.07 

1.09 

1.11 

1.14 

1.16 

1.19 

1.21 

1.24 

1.27 

1.31 

1.34 

43 

48 

1.00 

1.02 

1.03 

1.05 

1.07 

1.09 

1.11 

1.13 

1.16 

1.18 

1.21 

1.23 

1.26 

1.30 

1.33 

1.36 

42 

49 

1.02 

1.03 

1.05 

1.07 

1.09 

1.11 

.13 

1.15 

1.17 

1.20 

1.23 

1.25 

1.28 

1.32 

1.35 

1.39 

41 

50 

1.03 

1.05 

1.06 

1.08 

1.10 

1.12 

.14 

1.17 

1.19 

1.22 

1.24 

1.27 

1.30 

1.34 

1.37 

1.41 

40 

51 

1.05 

1.06 

1.08 

1.10 

1.12 

1.14 

.16 

1.18 

1.21 

1.23 

1.26 

1.29 

1.32 

1.35 

1.39 

1.43 

39 

52 

1.06 

1.08 

1.10 

1.11 

1.13 

1.15 

.18 

1.20 

1.23 

1.25 

1.28 

1.31 

1.34 

1.37 

1.41 

1.45 

38 

53 

1.07 

1.09 

1.11 

1.13 

1.15 

1.17 

.19 

1.22 

1.24 

1.27 

1.30 

1.33 

1.36 

1.39 

1.43 

1.47 

37 

54 

1.09 

1.11 

1.12 

1.14 

1.16 

1.19 

.21 

1.23 

1.26 

1.29 

1.31 

1.34 

1.38 

1.41 

1.45 

1.49 

36 

55 

1.10 

1  12 

1  14 

1  16 

1.18 

1.20 

.22 

1.25 

1.27 

1.30 

1.33 

1.36 

1.39 

1.43 

1.46 

1.50 

35 

56 

1.12 

1.13 

1.15 

1.17 

1.19 

1.22 

.24 

1.26 

1.29 

1.32 

1.35 

1.38 

1.41 

1.45 

1.48 

1.52 

34 

57 

1.13 

1.15 

1.17 

1.19 

1.21 

1.23 

.25 

1.28 

1.31 

1.33 

1.36 

.39 

1.43 

1.46 

1.50 

1.54 

33 

58 

1.14 

1.16 

1.18 

1.20 

1.22 

1.24 

.27 

1.29 

1.32 

1.35 

1.38 

.41 

1.44 

1.48 

1.52 

1.56 

32 

59 

1.15 

1.17 

1.19 

1.21 

1.23 

1.26 

.28 

1.31 

1.33 

1.36 

1.39 

.42 

1.46 

1.49 

1.53 

1.57 

31 

60 

1.17 

1.18 

1.20 

1.22 

1.25 

1.27 

.29 

1.32 

1.35 

1.38 

1.41 

.44 

1.47 

1.51 

1.55 

1.59 

30 

61 

1.18 

1.20 

1.22 

1.24 

1.26 

1.28 

.31 

1.33 

1.36 

1.39 

1.42 

.45 

1.49 

1.53 

1.56 

.61 

29 

63 

1.19 

1.21 

1.23 

1.25 

1.27 

1.29 

.32 

1.35 

1.37 

1.40 

1.43 

.47 

1.50 

1.54 

1.58 

.62 

28 

63 

1.20 

1.22 

1.24 

1.26 

1.28 

1.31 

.33 

1.36 

1.39 

1.42 

1.45 

.48 

1.52 

1.55 

1.59 

.64 

27 

64 

1.21 

1.23 

1.25 

1.27 

1.29 

1.32 

1.34 

1.37 

1.40 

1.43 

1.46 

.49 

1.53 

1.57 

1.61 

.65 

26 

65 

1.22 

1.24 

1.26 

1.28 

1.30 

1.33 

1.35 

1.38 

1.41 

1.44 

1.47 

.51 

1.54 

1.58 

1.62 

.66 

25 

66 

1.23 

1.25 

1.27 

1.29 

1.32 

1.34 

1.37 

1.39 

1.42 

.45 

1.48 

1.52 

1.55 

1.59 

1.63 

.68 

24 

67 

1.24 

1.26 

1.28 

1.30 

1.33 

1.35 

1.38 

1.40 

1.43 

.46 

1.50 

1.53 

1.57 

1.60 

1.65 

.69 

23 

68 

1.25 

1.27 

1.29 

1.31 

1.33 

1.36 

1.39 

1.41 

1.44 

.47 

1.51 

1.54 

1.58 

1.62 

1.66 

.70 

22 

69 

1.26 

1.28 

1.30 

1.32 

1.34 

1.37 

1.40 

1.42 

1.45 

.48 

1.52 

1.55 

1.59 

1.63 

1.67 

1.71 

21 

70 

1.26 

1.28 

1.31 

1.33 

1.35 

1.38 

1.40 

1.43 

1.46 

.49 

1.53 

1.56 

1.60 

1.64 

1.68 

1.73 

20 

71 

1.27 

1.29 

1.31 

1.34 

1.36 

1.39 

1.41 

1.44 

1.47 

.50 

1.54 

1.57 

1.61 

1.65 

1.69 

1.74 

19 

72 

1.28 

1.30 

1.32 

1.34 

1.37 

1.39 

1.42 

1.45 

1.48 

.51 

1.54 

1.58 

1.62 

1.66 

1.70 

1.75 

18 

73 

1.29 

1.31 

1.33 

1.35 

1.38 

1.40 

1.43 

1.46 

1.49 

.52 

1.55 

1.59 

1.63 

1.67 

1.71 

1.76 

17 

74 

1.29 

1.31 

1.34 

1.36 

1.38 

1.41 

1.44 

1.46 

1.49 

.53 

1.56 

1.60 

1.63 

1.68 

1.72 

1.76 

16 

75 

1.30 

1.32 

1.34 

1.37 

1.39 

1.42 

1.44 

1.47 

1.50 

.53 

1.57 

1.60 

1.64 

1.68 

1.73 

1.77 

15 

76 

1.31 

1.33 

1.35 

1.37 

1.40 

1.42 

1.45 

1.48 

1.51 

1.54 

1.58 

1.61 

1.65 

1.69 

.73 

.78 

14 

77 

1.31 

1.33 

1.35 

1.38 

1.40 

.43 

1.46 

.48 

1.52 

1.55 

1.58 

1.62 

1.66 

1.70 

.74 

.79 

13 

78 

1.32 

1.34 

1.36 

1.38 

1.41 

.43 

1.46 

.49 

1.52 

'  1.55 

1.59 

1.62 

1.66 

1.70 

.75 

.80 

12 

79 

1.32 

1.34 

1.36 

1.39 

1.41 

.44 

1.47 

.50 

1.53 

1.56 

1.59 

1.63 

1.67 

1.71 

.76 

.80 

11 

80 

1.33 

1.35 

1.37 

1.39 

1.42 

.44 

1.47 

.50 

1.53 

1.56 

1.60 

1.64 

1.87 

1.72 

.76 

.81 

10 

81 

1.33 

1.35 

.37 

1.40 

1.42 

.45 

.48 

.51 

1.54 

1.57 

1.60 

1.64 

1.68 

1.72 

.77 

.81 

9 

82 

1.33 

1.35 

.38 

1.40 

1.43 

.45 

.48 

.51 

1.54 

1.57 

1.61 

1.64 

1.68 

1.73 

.77 

.82 

8 

83 

1.34 

1.36 

.38 

1.40 

1.43 

.46 

.48 

.51 

1.54 

1.58 

1.61 

1.65 

1.69 

1.73 

.77 

.82 

7 

84 

1.34 

1.36 

.38 

1.41 

1.43 

.46 

.49 

.52 

1.55 

1.58 

1.62 

1.65 

1.69 

1.73 

.78 

.83 

6 

85 

1.34 

1.36 

.38 

1.41 

1.43 

.46 

.49 

.52 

1.55 

1.58 

1.62 

1.65 

1.69 

1.74 

.78 

.83 

5 

86 

1.34 

1.36 

1.39 

1.41 

1.44 

1.46 

.49 

1.52 

1.55 

1.59 

1.62 

1.66 

1.70 

1.74 

1.78 

.83 

4 

87 

1.34 

1.37 

1.39 

1.41 

1.44 

1.46 

.49 

1.52 

1.55 

1.59 

1.62 

1.66 

1.70 

1.74 

1.79 

.83 

3 

88 

1.34 

1.37 

1.39 

1.41 

1.44 

1.46 

.49 

1.52 

1.55 

1.59 

1.62 

1.66 

1.70 

1.74 

1.79 

.83 

2 

89 

1.35 

1.37 

1.39 

1.41 

1.44 

1.47 

1.49 

1.52 

1.56 

1.59 

1.62 

1.66 

1.70 

1.74 

1.79 

.84 

1 

90 

1.35 

1.37 

1.39 

1.41 

1.44 

1.47 

1.49 

1.52 

1.56 

1.59 

1.62 

1.66 

1.70 

1.74 

1.79 

.84 

0 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

50° 

51° 

52° 

53° 

54° 

55° 

56" 

57° 

8136°— 13 5 


66 


U.   S.   COAST   AND   GEODETIC   SURVEY    SPECIAL   PUBLICATION    NO.   14. 


Table  of  factors  for  reduction  of  transit  observations. 

TOP  ARGUMENT- STAR'S  DECLINATION  (S). 
SIDE  ARGUMENT- STAR'S  ZENITH  DISTANCE   (C). 
[For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  opposite  page,  j 


C 

57° 

58° 

58J° 

59° 

595° 

60° 

60  J° 

61° 

61J° 

62° 

62j° 

63° 

63J° 

64° 

645° 

65° 

C 

0 

1 

.03 

.03 

.03 

.03 

.03 

.03 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

89 

2 

.06 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.08 

.08 

.08 

.08 

.08 

.08 

88 

3 

.10 

.10 

.10 

.10 

.10 

.10 

.11 

.11 

.11 

.11 

.11 

.12 

.12 

.12 

.12 

.12 

87 

4 

.13 

.13 

.13 

.14 

.14 

.14 

.14 

.14 

.15 

.15 

.15 

.15 

.16 

.16 

.16 

.17 

86 

5 

.16 

.16 

.17 

.17 

.17 

.17 

.18 

.18 

.18 

.19 

.19 

.19 

.19 

.20 

.20 

.21 

85 

6 

.19 

.20 

.20 

.20 

.21 

.21 

.21 

.22 

.22 

.22 

.23 

.23 

.23 

.24 

.24 

.25 

84 

7 

.22 

.23 

.23 

.24 

.24 

.24 

.25 

.25 

.26 

.26 

.26 

.27 

.27 

.28 

.28 

.29 

83 

8 

.26 

.26 

.27 

.27 

.27 

.28 

.28 

.29 

.29 

.30 

.30 

.31 

.31 

.32 

.32 

.33 

82 

9 

.29 

.29 

.30 

.30 

.31 

.31 

.32 

.32 

.33 

.33 

.34 

.35 

.35 

.36 

.36 

.37 

81 

10 

.32 

.33 

.33 

.34 

.34 

.35 

.35 

.36 

.36 

.37 

.38 

.38 

.39 

.40 

.40 

.41 

80 

11 

.35 

.36 

.36 

.37 

.38 

.38 

.39 

.39 

.40 

..41 

.41 

.42 

.43 

.44 

.44 

.45 

79 

12 

.38 

.39 

.40 

.40 

.41 

.42 

.42 

.43 

.44 

.44 

.45 

.46 

.47 

.47 

.48 

.49 

78 

13 

.41 

.42 

.43 

.44 

.44 

.45 

.46 

.46 

.47 

.48 

.49 

.50 

.50 

.51 

.52 

.53 

77 

14 

.44 

.46 

.46 

.47 

.48 

.48 

.49 

.50 

.51 

.52 

.52 

.53 

.54 

.55 

.56 

,57 

76 

15 

.48 

.49 

.50 

.50 

.51 

.52 

.53 

.53 

.54 

.55 

.56 

.57 

.58 

.59 

.60 

.61 

75 

16 

.51 

.52 

.53 

.54 

.54 

.55 

.56 

.57 

.58 

.59 

.60 

.61 

.62 

.63 

.64 

.65 

74 

17 

.54 

.55 

.56 

.57 

.58 

.58 

.59 

.60 

.61 

.62 

.63 

.64 

.66 

.67 

.68 

.69 

73 

18 

.57 

.58 

.59 

.60 

.61 

.62 

.63 

.64 

.65 

.66 

.67 

.68 

.69 

.70 

.72 

.73 

72 

19 

.60 

.61 

.62 

.63 

.64 

.65 

.66 

.67 

.68 

.69 

.70 

.72 

.73 

.74 

.76 

.77 

71 

20 

.63 

.64 

.65 

.66 

.67 

.68 

.69 

.70 

.72 

.73 

.74 

.75 

.77 

.78 

.79 

.81 

70 

21 

.66 

.68 

.69 

.70 

.71 

.72 

.73 

.74 

.75 

.76 

.78 

.79 

.80 

.82 

.83 

.85 

69 

22 

.69 

.71 

.72 

.73 

.74 

.75 

.76 

.77 

.78 

.80 

.81 

.82 

.84 

.85 

.87 

.89 

68 

23 

.72 

.74 

.75 

.76 

.77 

.78 

.79 

.81 

.82 

.83 

.85 

.86 

.88 

.89 

.91 

.92 

67 

24 

.75 

.77 

.78 

.79 

.80 

.81 

.83 

.84 

.85 

.87 

.88 

.90 

.91 

.93 

.94 

.96 

66 

25 

.78 

.80 

.81 

.82 

.83 

.85 

.86 

.87 

.89 

.90 

.92 

.93 

.95 

.96 

.98 

1.00 

65 

26 

.80 

.83 

.84 

.85 

.86 

.88 

.89 

.90 

.92 

.93 

.95 

.97 

.98 

1.00 

1.02 

1.04 

64 

27 

.83 

.86 

.87 

.88 

.89 

.91 

.92 

.94 

.95 

.97 

.98 

1.00 

1.02 

1.04 

1.05 

1.07 

63 

28 

.86 

.89 

.90 

.91 

.93 

.94 

.95 

.97 

.98 

1.00 

1.02 

1.03 

1.05 

1.07 

1.09 

1.11 

62 

29 

.89 

.91 

.93 

.94 

.96 

.97 

.98 

1.00 

1.02 

1.03 

1.05 

1.07 

1.09 

1.11 

1.13 

1.15 

61 

30 

.92 

.94 

.96 

.97 

.99 

1.00 

1.01 

1.03 

1.05 

1.07 

1.08 

1.10 

1.12 

1.14 

1.16 

1.18 

60 

31 

.95 

.97 

.99 

1.00 

1.01 

1.03 

.05 

1.06 

1.08 

1.10 

1.11 

.13 

1.15 

1.17 

1.20 

1.22 

59 

32 

.97 

1.00 

1.01 

1.03 

1.04 

1.06 

.08 

1.09 

1.11 

1.13 

1.15 

.17 

1.19 

1.21 

1.23 

1.25 

58 

33 

1.00 

.03 

1.04 

1.06 

1.07 

1.09 

.11 

1.12 

1.14 

1.16 

1.18 

.20 

1.22 

1.24 

1.26 

1.29 

57 

34 

1.03 

.05 

1.07 

1.09 

1.10 

1.12 

.14 

1.15 

1.17 

1.19 

1.21 

.23 

1.25 

1.27 

1.30 

1.32 

56 

35 

1.05 

.08 

1.10 

1.11 

1.13 

1.15 

.16 

1.18 

1.20 

1.22 

1.24 

.26 

1.29 

1.31 

1.33 

1.36 

55 

36 

1.08 

.11 

1.12 

1.14 

1.16 

1.18 

.19 

1.21 

1.23 

1.25 

1.27 

.30 

1.32 

1.34 

1.37 

1.39 

54 

37 

1.10 

.14 

1.15 

1.17 

1.19 

1.20 

.22 

1.24 

1.26 

1.28 

1.30 

.33 

1.35 

1.37 

1.40 

1.42 

53 

38 

1.13 

.16 

1.18 

1.20 

1.21 

1.23 

1.25 

1.27 

1.29 

1.31 

1.33 

1.36 

1.38 

1.40 

1.43 

1.46 

52 

39 

1.15 

.19 

1.20 

1.22 

1.24 

1.26 

1.28 

1.30 

1.32 

1.34 

1.36 

1.39 

1.41 

1.43 

1.46 

1.49 

51 

40 

1.18 

.21' 

1.23 

1.25 

1.27 

1.29 

1.31 

1.33 

1.35 

1.37 

1.39 

1.42 

1.44 

1.47 

1.49 

1.52 

50 

41 

.20 

1.24 

1.26 

1.27 

1.29 

1.31 

1.33 

1.35 

1.37 

1.40 

1.42 

1.45 

1.17 

1.50 

1.52 

1.55 

49 

42 

.23 

1.26 

1.28 

1.30 

1.32 

1.34 

1.36 

1.38 

1.40 

1.42 

1.45 

1.47 

1.50 

1.53 

1.55 

1.58 

48 

43 

.25 

1.29 

1.30 

1.32 

1.34 

1.36 

1.39 

1.41 

1.43 

1.45 

1.48 

1.50 

1.53 

1.56 

1.58 

1.61 

47 

44 

.28 

1.31 

1.33 

1.35 

1.37 

1.39 

1.41 

1.43 

1.46 

1.48 

1.50 

1.53 

1.56 

1.58 

1.61 

1.64 

46 

45 

.30 

1.33 

1.35 

1.37 

1.39 

1.41 

1.44 

1.46 

1.48 

1.51 

1.53 

1.56 

1.58 

1.61 

1.64 

1.67 

45 

57° 

58° 

58}° 

59° 

59J" 

60° 

«0j° 

61" 

615° 

62° 

62J° 

63° 

635° 

64° 

64J° 

65° 

DETERMINATION    OP    TIME. 

Table  of  factors  for  reduction  of  transit  observations. 

TOP  ARGUMENT=STAR'S  DECLINATION  (J). 

SIDE  ARGUMENT- STAR'S  ZENITH  DISTANCE  (C). 

[For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  thi*  page.] 


67 


C 

57° 

58° 

58J° 

59° 

59j° 

60° 

60i° 

61° 

61J° 

62° 

62j° 

63° 

63J° 

64° 

64{° 

65° 

C 

0 

0 

46 

1.32 

1.36 

1.38 

1.40 

.42 

1.44 

1.46 

1.48 

.51 

1.53 

1.58 

1.58 

1.61 

1.64 

1.67 

1.70 

44 

47 

1.34 

1.38 

1.40 

1.42 

.44 

1.46 

1.49 

1.51 

.53 

1.56 

1.58 

1.61 

1.64 

1.67 

1.70 

1.73 

43 

48 

1.36 

1.40 

1.42 

1.44 

.46 

1.48 

1.51 

1.53 

.55 

.58 

1.60 

1.53 

1.66 

1.69 

1.72 

1.76 

42 

49 

1.39 

1.42 

1.44 

1.47 

.49 

1.51 

1.53 

1.56 

.58 

.61 

.K 

1.66 

.69 

1.72 

1.75 

1.79 

41 

50 

1.41 

1.44 

1.47 

1.49 

.51 

1.53 

1.56 

1.58 

.60 

.63 

.66 

1.69 

.72 

1.75 

1.78 

1.81 

40 

51 

1.43 

1.47 

1.49 

1.51 

.53 

1.55 

1.58 

1.60 

.63 

.66 

.68 

1.71 

.74 

1.77 

1.80 

1.84 

39 

52 

1.45 

1.49 

1.51 

1.53 

.55 

1.58 

1.60 

1.63 

.65 

.68 

.71 

1.74 

.77 

1.80 

1.83 

1.86 

38 

53 

1.47 

1.51 

1.53 

1.55 

.57 

1.60 

1.62 

1.65 

.67 

.70 

.73 

1.76 

.79 

1.82 

1.85 

1.89 

37 

54 

1.49 

1.53 

1.55 

1.57 

.59 

1.62 

1.64 

1.67 

1.69 

.72 

1.75 

1.78 

.81 

1.85 

1.88 

1.91 

36 

55 

1.50 

1.55 

1.57 

1.59 

.61 

1.64 

1.66 

1.69 

1.72 

1.74 

1.77 

1.80 

.84 

1.87 

1.90 

1.94 

35 

56 

1.52 

1.56 

1.59 

1.61 

.63 

1.66 

1.68 

1.71 

1.74 

1.77 

1.80 

1.83 

1.86 

1.89 

1.93 

1.96 

34 

57 

1.54 

1.58 

1.61 

1.63 

.65 

1.68 

1.70 

1.73 

1.76 

1.79 

1.82 

1.85 

1.88 

1.91 

1.95 

1.98 

33 

58 

1.56 

1.60 

1.62 

1.65 

.67 

1.70 

1.72 

1.75 

1.78 

1.81 

1.84 

1.87 

1.90 

1.93 

1.97 

2.01 

32 

59 

1.57 

1.62 

1.64 

1.66 

.69 

1.71 

.74 

1.77 

1.80 

1.83 

1.86 

1.89 

1.92 

1.96 

1  99 

2  03 

31 

60 

1.59 

1.63 

1.66 

1.68 

.71 

1.73 

.76 

1.79 

1.81 

1.84 

1.88 

1.91 

1.94 

1.98 

2.01 

2.05 

30 

61 

1.61 

1.65 

1.67 

1.70 

.72 

1.75 

.78 

1.80 

.83 

1.86 

1.89 

1.93 

1.96 

2.00 

2.03 

2.07 

29 

62 

1.62 

1.67 

1.69 

1.71 

.74 

1.77 

.79 

1.82 

.85 

1.88 

1.91 

1.94 

1.98 

2.01 

2.05 

2.09 

28 

63 

1.64 

1.68 

1.70 

1.73 

.76 

1.78 

.81 

1.84 

.87 

1.90 

1.93 

1.96 

2.00 

2.03 

2.07 

2.11 

27 

64 

1.65 

1.70 

1.72 

1.75 

.77 

1.80 

.83 

1.85 

.88 

1.91 

1.95 

1.98 

2.02 

2.05 

2.09 

2.13 

26 

65 

1.66 

1.71 

1.73 

1.76 

.79 

1.81 

.84 

1.87 

.90 

1.93 

1.96 

2.00 

2.03 

2.07 

2.11 

2.14 

25 

66 

1.68 

1.72 

1.75 

1.77 

.80 

1.83 

.85 

1.88 

.91 

1.95 

1.98 

2.01 

2.05 

2.08 

2.12 

2.16 

24 

67 

1.69 

1.74 

1.76. 

1.79 

.81 

1.84 

.87 

1.90 

.93 

1.96 

1.99 

2.03 

2.06 

2.10 

2.14 

2.18 

23 

68 

1.70 

1.75 

1.77 

1.80 

.83 

1.85 

.88 

1.91 

.94 

1.97 

2.01 

2.04 

2.08 

2.11 

2.15 

2.19 

22 

69 

1.71 

1.76 

1.79 

1.81 

1.84 

1.87 

.90 

1.93 

.96 

1.99 

2.02 

2.06 

2.09 

2.13 

2.17 

2.21 

21 

70 

1.73 

1.77 

1.80 

1.82 

1.85 

1.88 

.91 

1.94 

.97 

2.00 

2.03 

2.07 

2.11 

2.14 

2.18 

2.22 

20 

71 

1.74 

1.78 

1.81 

1.84 

1.86 

1.89 

.92 

1.95 

.98 

2.01 

2.05 

2.08 

2.12 

2.16 

2.20 

2.24 

19 

72 

1.75 

1.79 

1.82 

1.85 

1.87 

1.90 

.93 

1.96 

.99 

2.03 

2.06 

2.09 

2.13 

2.17 

2.21 

2.25 

18 

73 

1.76 

1.80 

1.83 

1.86 

1.88 

1.91 

.94 

1.97 

2.00 

2.04 

2.07 

2.11 

2.14 

2.18 

2.22 

2.26 

17 

74 

1.76 

1.81 

1.84 

1.87 

1.89 

1.92 

1.95 

1.98 

2.01 

2.05 

2.08 

2.12 

2.15 

2.19 

2.23 

2.27 

16 

75 

1.77 

1.82 

1.85 

1.88 

1.90 

1.93 

1.96 

1.99 

2.02 

2.06 

2.09 

2.13 

2.16 

2.20 

2.24 

2.29 

15 

76 

1.78 

1.83 

1.86 

1.88 

1.91 

1.94 

1.97 

2.00 

2.03 

2.07 

2.10 

2.14 

2.17 

2.21 

2.25 

2.30 

14 

77 

1.79 

1.84 

1.86 

1.89 

1.92 

1.95 

1.98 

2.01 

2.04 

2.07 

2.11 

2.15 

2.18 

2.22 

2.26 

2.31 

13 

78 

1.80 

1.85 

1.87 

1.90 

1.93 

1.96 

1.99 

2.02 

2.05 

2.08 

2.12 

2.15 

2.19 

2.23 

2.27 

2.31 

12 

79 

1.80 

1.85 

1.88 

1.91 

1.93 

1.96 

1.99 

2.02 

2.06 

2.09 

2.13 

2.16 

2.20 

2.24 

2.28 

2.32 

11 

80 

1.81 

1.86 

1.88 

1.91 

1.94 

1.97 

2.00 

2.03 

2.06 

2.10 

2.13 

2.17 

2.21 

2.25 

2.29 

2.33 

10 

81 

1.81 

1.86 

1.89 

1.92 

1.95 

1.98 

2.01 

2.04 

2.07 

2.10 

2.14 

2.18 

2.21 

2.25 

2.29 

2.34 

9 

82 

1.82 

1.87 

1.90 

1.92 

1.95 

1.98 

2.01 

2.04 

2.08 

2.11 

2.15 

2.18 

2.22 

2.26 

2.30 

2.34 

8 

83 

1.82 

1.87 

1.90 

1.93 

1.96 

1.99 

2.02 

2.05 

2.08 

2.12 

2.15 

2.19 

2.22 

2.26 

2.31 

2.35 

7 

84 

1.83 

1.88 

1.90 

1.93 

1.96 

1.99 

2.02 

2.05 

2.08 

2.12 

2.15 

2.19 

2.23 

2.27 

2.31 

2.35 

6 

85 

1.83 

1.88 

1.91 

1.93 

1.96 

1.99 

2.02 

2.05 

2.09 

2.12 

2.16 

2.19 

2.23 

2.27 

2.31 

2.36 

5 

86 

1.83 

1.88 

1.91 

1.94 

1.97 

2.00 

2.03 

2.06 

2.09 

2.13 

2.16 

2.20 

2.24 

2.28 

2.32 

2.36 

4 

NT 

1.83 

1.88 

1.91 

1.94 

1.97 

2.00 

2.03 

2.06 

2.09 

2.13 

2.16 

2.20 

2.24 

2.28 

2.32 

2.36 

3 

\S 

1.83 

1.89 

1.91 

1.94 

1.97 

2.00 

2.03 

2.06 

2.09 

2.13 

2.16 

2.20 

2.24 

2.28 

2.32 

2.36 

2 

89 

1.84 

1.89 

1.91 

1.94 

1.97 

2.00 

2.03 

2.06 

2.10 

2.13 

2.17 

2.20 

2.24 

2.28 

2.32 

2.37 

1 

!M> 

1.84 

1.89 

1.91 

1.94 

1.97 

2.00 

2.03 

2.06 

2.10 

2.13 

2.17 

2.20 

2.24 

2.28 

2.32 

2.37 

0 

57° 

58° 

58J" 

59° 

59J° 

60° 

60J° 

61° 

81}« 

62° 

62J° 

63° 

63}° 

64° 

64J» 

65" 

68 


U.   S.   COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 


Table  of  factors  for  reduction  of  transit  observations. 

TOP  ARGUMENT=STAR'S  DECLINATION  (3). 

SIDE  ARGUMENT=STAR'S  ZENITH  DISTANCE  (C). 

[For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  opposite  page.] 


C 

65" 

65}° 

66° 

66}° 

67° 

67J° 

68° 

681° 

69° 

69°  10' 

69°  20' 

69°  30' 

69°  40' 

69°  50' 

70° 

70°  10' 

; 

1 

.04 

.04 

.04 

.04 

.04 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

89 

2 

.08 

.08 

.09 

.09 

.09 

.09 

.09 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

.10 

88 

3 

.12 

.13 

.13 

.13 

.13 

.14 

.14 

.14 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

.15 

87 

4 

.17 

.17 

.17 

.18 

.18 

.18 

.19 

.19 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

.20 

86 

5 

.21 

.21 

.21 

.22 

.22 

.23 

.23 

.24 

.24 

.24 

.25 

.25 

.25 

.25 

.25 

.26 

85 

6 

.25 

.25 

.26 

.26 

.27 

.27 

.28 

.28 

.29 

.29 

.30 

.30 

.30 

.30 

.31 

.31 

84 

7 

.29 

.29 

.30 

.31 

.31 

.32 

.33 

.33 

.34 

.34 

.34 

.35 

.35 

.35 

.36 

.36 

83 

8 

.33 

.34 

.34 

.35 

.36 

.36 

.37 

.38 

.39 

.39 

.39 

.40 

.40 

.40 

.41 

.41 

82 

9 

.37 

.38 

.39 

.39 

.40 

.41 

.42 

.43 

.44 

.44 

.44 

.45 

.45 

.45 

.46 

.46 

81 

10 

.41 

.42 

.43 

.43 

.44 

.45 

.46 

.47 

.48 

.49 

.49 

.50 

.50 

.50 

.51 

.51 

80 

11 

.45 

.46 

.47 

.48 

.49 

.50 

.51 

.52 

.53 

.54 

.54 

.54 

.55 

.55 

.56 

.56 

79 

12 

.49 

.50 

.51 

.52 

.53 

.54 

.56 

.57 

.58 

.58 

.59 

.59 

.60 

.60 

.61 

.61 

78 

13 

.53 

.54 

.55 

.56 

.58 

.59 

.60 

.61 

.63 

.63 

.64 

.64 

.65 

.65 

.66 

.66 

77 

14 

.57 

.58 

.59 

.61 

.62 

.63 

.65 

.66 

.67 

.68 

.68 

.69 

.70 

.70 

.71 

.71 

76 

15 

.61 

.62 

.64 

.65 

.66 

.68 

.69 

.71 

.72 

.73 

.73 

.74 

.74 

.75 

.76 

.76 

75 

16 

.65 

.66 

.68 

.60 

.71 

.72 

.74 

.75 

.77 

.78 

.78 

.79 

.79 

.80 

.81 

.81 

74 

17 

.69 

.70 

.72 

.73 

.75 

.76 

.78 

.80 

.81 

.82 

.83 

.83 

.84 

.85 

.85 

.86 

73 

18 

.73 

.74 

.76 

.77 

.79 

.81 

.83 

.84 

.86 

.87 

.88 

.88 

.89 

.90 

.90 

.91 

72 

19 

.77 

.78 

.80 

.82 

.83 

.85 

.87 

.89 

.91 

.92 

.92 

.93 

.94 

.94 

.95 

.96 

71 

20 

.81 

.82 

.84 

.86 

.88 

.89 

.91 

.93 

.95 

.96 

.97 

.98 

.98 

.99 

1.00 

1.01 

70 

21 

.85 

.86 

.88 

.90 

.92 

.94 

.96 

.98 

1.00 

.01 

1.02 

.02 

1.03 

1.04 

1.05 

1.06 

69 

22 

.89 

.90 

.92 

.94 

.96 

.98 

1.00 

1.02 

1.05 

.05 

1.06 

.07 

1.08 

.  1.09 

1.09 

1.10 

68 

23 

.92 

.94 

.96 

.98 

1.00 

1.02 

1.04 

1.07 

1.09 

.10 

1.11 

.12 

1.12 

'  1.13 

1.14 

1.15 

67 

24 

.96 

.98 

1.00 

1.02 

1.04 

1.06 

1.09 

1.11 

1.14 

.14 

1.15 

.16 

1.17 

1.18 

1.19 

1.20 

66 

25 

1.00 

1.02 

1.04 

1.06 

1.08 

1.10 

1.13 

1.15 

1.18 

.19 

1.20 

.21 

1.22 

1.23 

1.24 

1.25 

65 

26 

1.04 

1.06 

1.08 

1.10 

1.12 

1.15 

1.17 

1.20 

1.22 

1.23 

.24 

.25 

1.26 

.27 

1.28 

1.29 

64 

27 

1.07 

1.09 

1.12 

1.14 

1.16 

1.19 

1.21 

1.24 

1.27 

1.28 

.29 

.30 

1.31 

.32 

1.33 

1.34 

63 

28 

1.11 

1.13 

1.15 

1.18 

1.20 

1.23 

1.25 

1.28 

1.31 

1.32 

.33 

.34 

1.35 

.36 

1.37 

1.38 

62 

29 

1.15 

1.17 

1.19 

.22 

1.24 

1.27 

1.29 

1.32 

1.35 

1.36 

.37 

.38 

1.40 

.41 

1.42 

1.43 

61 

30 

1.18 

1.21 

1.23 

.25 

1.28 

1.31 

1.33 

1.36 

1.39 

1.41 

.42 

.43 

1.44 

.45 

1.46 

1.47 

60 

31 

1.22 

1.24 

.27 

.29 

1.32 

1.35 

1.38 

1.40 

1.44 

1.45 

.46 

.47 

1.48 

.49 

1.51 

1.52 

59 

32 

1.25 

1.28 

.30 

.33 

1.36 

1.39 

1.42 

.45 

1.48 

1.49 

.50 

.51 

1.52 

.54 

1.55 

1.56 

58 

33 

1.29 

1.31 

.34 

.37 

1.39 

1.42 

1.45 

.49 

1.52 

1.53 

.54 

.55 

1.57 

.58 

1.59 

1.60 

57 

34 

1.32 

1.35 

.37 

.40 

1.43 

1.46 

1.49 

.53 

1.56 

1.57 

.58 

.60 

1.61 

.62 

1.63 

1.65 

56 

35 

1.36 

1.38 

.41 

.44 

1.47 

1.50 

1.53 

.56 

1.60 

1.61 

.62 

1.64 

1.65 

.66 

1.68 

1.69 

55 

36 

1.39 

1.42 

.45 

.47 

1.51 

1.54 

1.57 

.60 

1.64 

1.55 

.66 

1.68 

1.69 

.70 

1.72 

1.73 

54 

37 

1.42 

1.45 

.48 

.51 

1.54 

1.57 

1.61 

.64 

1.68 

1.69 

.70 

1.72 

1.73 

.74 

1.76 

1.77 

53 

38 

1.46 

1.48 

.51 

.54 

1.58 

1.61 

1.64 

.68 

1.72 

1.73 

.74 

1.76 

1.77 

.79 

1.80 

1.82 

52 

39 

1.49 

1.52 

.55 

.58 

1.61 

1.65 

1.68 

.72 

1.75 

1.77 

.78 

1.80 

1.81 

.82 

1.84 

1.86 

51 

40 

1.52 

1.55 

.58 

.61 

1.65 

1.68 

1.72 

.75 

1.79 

1.81 

.82 

1.84 

1.85 

.86 

1.88 

1.89 

50 

41 

1.55 

1.58 

.61 

.64 

1.68 

.71 

1.75 

.79 

1.83 

1.84 

.86 

1.87 

1.89 

.90 

1.92 

1.93 

49 

42 

1.58 

1.61 

.64 

.68 

1.71 

.75 

1.79 

.83 

1.87 

1.88 

.90 

1.91 

1.93 

.94 

1.96 

1.97 

48 

43 

1.61 

1.64 

.68 

.71 

1.75 

.78 

1.82 

.86 

1.90 

1.92 

.93 

1.95 

1.96 

.98 

1.99 

2.01 

47 

44 

1.64 

1.67 

.71 

.74 

1.78 

.82 

1.85 

1.90 

1.94 

1.95 

.97 

1.98 

2.00 

2.02 

2.03 

2.05 

46 

45 

1.67 

1.70 

.74 

1.77 

1.81 

.85 

1.89 

1.93 

1.97 

1.99 

2.00 

2.02 

2.04 

2.05 

2.07 

2.08 

45 

65° 

65}° 

66° 

661° 

67° 

671° 

68° 

68}° 

69° 

69°  10' 

69°  20' 

69°  30' 

69°  40' 

69°  50' 

70° 

70°  10' 

DETERMINATION   OF   TIME. 


69 


Table  of  factors  for  reduction  of  transit  observations. 

TOP  ARGUMENT- STAR'S  DECLINATION  (d). 
SIDE  ARGUMENT=STAR'S  ZENITH  DISTANCE  (0 
[  For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  this  page.] 


C 

65° 

65J° 

66° 

66*° 

67° 

67  j° 

68° 

68J° 

69° 

69°  10' 

69°  20' 

69°  30* 

69°  40' 

69°  50' 

70° 

70°  10' 

C 

46 

1.70 

1.74 

1.77 

1.80 

1.84 

1.88 

1.92 

1.96 

2.01 

2.02 

2.04 

2.05 

2.07 

2.09 

2.10 

2.12 

44 

47 

1.73 

1.76 

1.80 

1.83 

1.87 

1.91 

1.95 

2.00 

2.04 

2.06 

2.07 

2.09 

2.10 

2.12 

2.14 

2.16 

43 

48 

1.76 

1.79 

1.83 

1.86 

1.90 

1.94    1.98 

2.03 

2.07 

2.09 

2.11 

2.12 

2.14 

2.16 

2.17 

2.19 

42 

49 

1.79 

1.82 

1.86 

1.89 

1.93 

1.97    2.01 

2.06 

2.11 

2.12 

2.14 

2.16 

2.17 

2.19 

2.21 

2.22 

41 

50 

1.81 

1.85 

1.88 

1.92 

1.96 

2.00    2.04 

2.09 

2.14 

2.15 

2.17 

2.19 

2.20 

2.22 

2.24 

2.26 

40 

51 

1.84 

1.87 

1.91 

1.95 

1.99 

2.03  i   2.07 

2.12 

2.17 

2.18 

2.20 

2.22 

2.24 

2.25 

2.27 

2.29 

39 

52 

1.86 

1.90 

1.94 

1.98 

2.02 

2.06 

2.10 

2.15 

2.20 

2.22 

2.23 

2.25 

2.27 

2.29 

2.30 

2.32 

38 

53 

1.89 

1.93 

1.96 

2.00 

2.04 

2.09 

2.13 

2.18 

2.23 

2.25 

2.26 

2.28 

2.30 

2.32 

2.33 

2.35 

37 

54 

1.91 

1.95 

1.99 

2.03 

2.07 

2.11 

2.16 

2.21 

2.26 

2.28 

'2.29 

2.31 

2.33 

2.35 

2.37 

2.38 

36 

55 

1.94 

1.98 

2.01 

2.05 

2.10 

2.14 

2.19 

2.23 

2.29 

2.30 

2.32 

2.34 

2.36 

2.38 

2.40 

2.41 

35 

56 

1.96 

2.00 

2.04 

2.08 

2.12 

2.17 

2.21 

2.26 

2.31 

2.33 

2.35 

2.37 

2.39 

2.40 

2.42 

2.44 

34 

57 

1.98 

2.02 

2.06 

2.10 

2.15 

2.19 

2.24 

2.29 

2.34 

2.36 

2.38 

2.39 

2.41 

2.43 

2.45 

2.47 

33 

58 

2.01 

2.05 

2.08 

2.13 

2.17 

2.22 

2.26 

2.31 

2.37 

2.38 

2.40 

2.42 

2.44 

2.46 

2.48 

2.50 

32 

59 

2.03 

2.07 

2.11 

2.15 

2.19 

2.24 

2.29 

2.34 

2.39 

2.41 

2.43 

2.45 

2.47 

2.49 

2.51 

2.53 

31 

60 

2.05 

2.09 

2.13 

2.17 

2.22 

2.26 

2.31    2.36 

2.42 

2.44 

2.45 

2.47 

2.49 

2.51 

2.53 

2.55 

30 

61 

2.07 

2.11 

2.15 

2.19 

2.24 

2.29 

2.33 

2.39 

2.44 

2.46 

2.48 

2.50 

2.52 

2.54 

2.56 

2.58 

29 

62 

2.09 

2.13 

2.17 

2.21 

2.26 

2.31 

2.36 

2.41 

2.46 

2.48 

2.50 

2.52 

2.54 

2.56 

2.58 

2.60 

28 

63 

2.11 

2.15 

2.19 

2.23 

2.28 

2.33 

2.38 

2.43 

2.49 

2.50 

2.52 

2.54 

2.56 

2.58 

2.60 

2.63 

27 

64 

2.13 

2.17 

2.21 

2.25 

2.30 

2.35  !   2.40 

2.45 

2.51 

2.53 

2.55 

2.57 

2.59 

2.61 

2.63 

2.65 

26 

65 

2.14 

2.19 

2.23 

2.27 

2.32 

2.37 

2.42 

2.47 

2.53 

2.55 

2.57 

2.59 

2.61 

2.63 

2.65 

2.67 

25 

66 

2.16 

2.20 

2.25 

2.29 

2.34 

2.39 

2.44 

2.49 

2.55 

2.57 

2.59 

2.61 

2.63 

2.65 

2.67 

2.69 

24 

67 

2.18 

2.22 

2.26 

2.31 

2.36 

2.41 

2.46 

2.51 

2.57 

2.59 

2.61 

2.63 

2.65 

2.67 

2.69 

2.71 

23 

68 

2.19 

2.24 

2.28 

2.32 

2.37 

2.42 

2.47 

2.53 

2.59 

2.61 

2.63 

2.65 

2.67 

2.69 

2.71 

2.73 

22 

69 

2.21 

2.25 

2.30 

2.34 

2.39 

2.44 

2.49 

2.55 

2.61 

2.62 

2.64 

2.67 

2.69 

2.71 

2.73 

2.75 

21 

JO 

2.22 

2.27 

2.31 

2.36 

2.40 

2.46 

2.51 

2.56 

2.62 

2.64 

2.66 

2.68 

2.70 

2.73 

2.75 

2.77 

20 

71 

2.24 

2.28 

2.32 

2.37 

2.42 

2.47 

2.52 

2.58 

2.64 

2.66 

2.68 

2.70 

2.72 

2.74 

2.77 

2.79 

19 

72 

2.25 

2.29 

2.34 

2.38 

2.43 

2.49 

2.54 

2.59 

2.65 

2.67 

2.70 

2.72 

2.74 

2.76 

2.78 

2.80 

18 

73 

2.26 

2.31 

2.35 

2.40 

2.45 

2.50 

2.55 

2.61 

2.67 

2.69 

2.71 

2.73 

2.75 

2.77 

2.80 

2.82 

17 

74 

2.27 

2.32 

2.36 

2.41 

2.46 

2.51 

2.57 

2.62 

2.68 

2.70 

2.72 

2.74 

2.77 

2.79 

2.81 

2.83 

16 

75 

2.29 

2.33 

2.37 

2.42 

2.47 

2.52 

2.58 

2.64 

2.70 

2.72 

2.74 

2.76 

2.78 

2.80 

2.82 

2.85 

15 

76 

2.30 

2.34 

2.39 

2.43 

2.48 

2.54 

2.59 

2.65 

2.71 

2.73 

2.75 

2.77 

2.79 

2.81 

2.84 

2.86 

14 

77 

2.31 

2.35 

2.40 

2.44 

2.49 

2.55 

2.60 

2.66 

2.72 

2.74 

2.76 

2.78 

2.80 

2.83 

2.85 

2.87 

13 

78 

2.31 

2.36 

2.40 

2.45 

2.50 

2.56 

2.61 

2.67 

2.73 

2.75 

2.77 

2.79 

2.81 

2.84 

2.86 

2.88 

12 

79 

2.32 

2.37 

2.41 

2.46 

2.51 

2.57 

2.62 

2.68 

2.74 

2.76 

2.78 

2.80 

2.82 

2.85 

2.87 

2.89 

11 

80 

2.33 

2.38 

2.42 

2.47 

2.52 

2.57 

2.63 

2.69 

2.75 

2.77 

2.79 

2.81 

2.83 

2.86 

2.88 

2.90 

10 

81 

2.34 

2.38 

2.43 

2.48 

2.53 

2.58 

2.64 

2.69 

2.76 

2.78 

2.80 

2.82 

2.84 

2.86 

2.89 

2.91 

9 

82 

2.34 

2.39 

2.43 

2.48 

2.53 

2.59 

2.64 

2.70 

2.76 

2.78 

2.81 

2.83 

2.85 

2.87 

2.90 

2.92 

8 

83 

2.35 

2.39 

2.44 

2.49 

2.54 

2.59 

2.65 

2.71 

2.77 

2.79 

2.81 

2.83 

2.86 

2.88 

2.90 

2.92 

7 

84 

2.35 

2.40 

2.45 

2.49 

2.55 

2.60 

2.66 

2.71 

2.78 

2.80 

2.82 

2.84 

2.86 

2.88 

2.91 

2.93 

6 

85 

2.36 

2.40 

2.45 

2.50 

2.56 

2.60 

2.66 

2.72 

2.78 

2.80 

2.82 

2.84 

2.87 

2.89 

2.91 

2.94 

5 

86 

2.36 

2.41 

2.45 

2.50 

2.55 

2.61 

2.66 

2.72 

2.78 

2.80 

2.83 

2.85 

2.87 

2.89 

2.92 

2.94 

4 

87 

2.36 

2.41 

2.46 

2.50 

2.56 

2.61 

2.67 

2.72 

2.79 

2.81 

2.83 

2.85 

2.87 

2.90 

2.92 

2.94 

3 

88 

2.36 

2.41 

2.46 

2.51 

2.56 

2.61 

2.67 

2.73 

2.79 

2.81 

2.83 

2.85 

2.88 

2.90 

2.92 

2.95 

2 

89 

2.37 

2.41 

2.46 

2.51 

2.56 

2.61 

2.67 

2.73 

2.79 

2.81 

2.83 

2.86 

2.88 

2.90 

2.92 

2.95 

1 

90 

2.37 

2.41 

2.46 

2.51 

2  56 

2.61 

2.67 

2.73 

2.79 

2.81 

2.83 

2.86 

2.88 

2.90 

2.92 

2.95 

0 

65° 

65j° 

66° 

66J° 

67° 

67j° 

68° 

esr 

69° 

69°  107 

69°  yy 

69°  30' 

69  °40' 

69°  50' 

70° 

70°  10* 

70 


TJ.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION   NO.   14. 


Table  of  factors  for  reduction  of  transit  observations. 

TOP  AROUMENT=STAR'S  DECLINATION  (a). 

SIDE  ARGUMENT-STAB'S  ZENITH  DISTANCE  (C). 

[For  factor  A  use  left-hand  argument.    For  factor  S  use  right-hand  argument.    For  factor  C  use  bottom  line  on  opposite  page.] 


C 

70°  10' 

70°  20' 

70°  30' 

70°  40' 

70°  50' 

71° 

71°  10' 

71°  20' 

71°  30' 

71°4CK 

71°  50' 

72° 

72°  10' 

72°  20' 

72°  30' 

72°  40' 

C 

o 

1 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

89 

2 

.10 

.10 

.10 

.10 

.11 

.11 

.11 

.11 

.11 

.11 

.11 

.11 

.11 

.12 

.12 

.12 

88 

3 

.15 

.16 

.16 

.16 

.16 

.16 

.16 

.18 

.16 

.17 

.17 

.17 

.17 

.17 

.17 

.18 

87 

4 

.20 

.21 

.21 

.21 

.21 

.21 

.22 

.22 

.22 

.22 

.22 

.23 

.23 

.23 

.23 

.23 

86 

5 

.26 

.26 

.26 

.26 

.26 

.27 

.27 

.27 

.27 

.28 

.28 

.28 

.28 

.29 

.29 

.29 

85 

6 

.31 

.31 

.31 

.32 

.32 

.32 

.32 

.33 

.33 

.33 

.34 

.34 

.34 

.34 

.35 

.35 

84 

7 

.36 

.36 

.37 

.37 

.37 

.37 

.38 

.38 

.38 

.39 

.39 

.39 

.40 

.40 

.41 

.41 

83 

8 

.41 

.41 

.42 

.42 

.42 

.43 

.43 

.44 

.44 

.44 

.45 

.45 

.45 

.46 

.46 

.47 

82 

9 

.46 

.46 

.47 

.47 

.48 

.48 

.48 

.49 

.49 

.50 

.50 

.51 

.51 

.52 

.52 

.52 

81 

10 

.51 

.52 

.52 

.52 

.53 

.53 

.54 

.54 

.55 

.55 

.56 

.56 

.57 

.57 

.58 

.58 

80 

11 

.56 

.57 

.57 

.58 

.58 

.59 

.59 

.60 

.60 

.61 

.61 

.62 

.62 

.63 

.63 

.64 

79 

12 

.61 

.62 

.62 

.63 

.63 

.64 

.64 

.65 

.66 

.66 

.67 

.67 

.68 

.68 

.69 

.70 

78 

13 

.66 

.67 

.67 

.68 

.68 

.69 

.70 

.70 

.71 

.72 

.72 

.73 

.74 

.74 

.75 

.76 

77 

14 

.71 

.72 

.72 

.73 

.74 

.74 

.75 

.76 

.76 

.77 

.78 

.78 

.79 

.80 

.80 

.81 

76 

15 

.76 

.77 

.78 

.78 

.79 

.79 

.80 

.81 

.81 

.82 

.83 

.84 

.84 

.85 

.86 

.87 

75 

16 

.81 

.82 

.83 

.83 

.84 

.85 

.85 

.86 

.87 

.88 

.88 

.89 

.90 

.91 

.92 

.92 

74 

17 

.86 

.87 

.88 

.88 

.89 

.90 

.90 

.91 

.92 

.93 

.94 

.95 

.96 

.96 

.97 

.98 

73 

18 

.91 

.92 

.93 

.93 

.94 

.95 

.96 

.96 

.97 

.98 

.99 

1.00 

1.01 

1.02 

1.03 

1.04 

72 

19 

.96 

.97 

.98 

.98 

.99 

1.00 

1.01 

1.02 

1.03 

1.04 

1.04 

1.05 

1.06 

1.07 

1.08 

1.09 

71 

20 

1.01 

1.02 

1.02 

1.03 

1.04 

1.05 

1.06 

1.07 

1.08 

1.09 

1.10 

1.11 

1.12 

1.13 

1.14 

1.15 

70 

21 

1.06 

.06 

1.07 

.08 

1.09 

1.10 

1.11 

1.12 

1.13 

1.14 

1.15 

1.16 

1.17 

1.18 

1.19 

1.20 

69 

22 

1.10 

.11 

1.12 

.13 

1.14 

1.15 

1.16 

1.17 

1.18 

1.19 

1.20 

1.21 

1.22 

1.24 

1.25 

1.26 

68 

23 

1.15 

.16 

1.17 

.18 

1.19 

1.20 

1.21 

1.22 

1.23 

1.24 

1.25 

1.26 

1.28 

1.29 

1.30 

1.31 

67 

24 

1.20 

.21 

1.22 

.23 

1.24 

1.25 

1.26 

1.27 

1.28 

1.29 

1.30 

1.32 

1.33 

1.34 

1.35 

1.36 

66 

25 

1.25 

.26 

1.27 

.28 

1.29 

1.30 

1.31 

1.32 

1.33 

1.34 

1.36 

1.37 

1.38 

1.39 

1.41 

1.42 

65 

26 

1.29 

.30 

1.31 

1.32 

1.34 

1.35 

1.36 

1.37 

1.38 

1.39 

1.41 

1.42 

1.43 

1.44 

1.46 

1.47 

64 

27 

1.34 

.35 

1.36 

1.37 

1.38 

1.39 

1.41 

1.42 

1.43 

1.44 

1.46 

1.47 

1.48 

1.50 

1.51 

1.52 

63 

28 

1.38 

.40 

1.41 

1.42 

1.43 

1.44 

1.45 

1.47 

1.48 

1.49 

1.51 

1.52 

1.53 

1.55 

1.56 

1.58 

62 

29 

1.43 

.44 

1.45 

1.46 

1.48 

1.49 

1.50 

1.52 

1.53 

1.54 

1.56 

1.57 

1.58 

1.60 

1.61 

1.63 

61 

30 

1.47 

.49 

1  50 

1  51 

1.52 

1.54 

1.55 

1.56 

1.58 

1.59 

1.60 

1.62 

1.63 

1.65 

1.66 

1.68 

60 

31 

1.52 

.53 

1.54 

1.56 

1.57 

1.S8 

1.60 

1.61 

1.62 

1.64 

1.65 

1.67 

1.68 

1.70 

1.71 

1.73 

59 

32 

1.56 

.57 

1.59 

1.60 

1.61 

1.63 

1.64 

1.66 

1.67 

1.68 

1.70 

1.71 

1.73 

1.75 

1.76 

1.78 

58 

33 

1.60 

.62 

1.63 

1.64 

1.66 

1.67 

1.69 

.70 

1.72 

1  73 

1  75 

1.76 

1.78 

1.80 

1.81 

1.83 

57 

34 

1.65 

.66 

1.68 

1.69 

1.70 

1.72 

1.73 

.75 

1.76 

1.78 

1.79 

1.81 

1.83 

1.84 

1.86 

1.88 

56 

35 

1.69 

.70 

1.72 

1.73 

1.75 

1.76 

1.78 

.79 

1.81 

1.82 

1.84 

1.86 

1.87 

1.89 

1.91 

1.92 

55 

36 

1.73 

.75 

1.76 

1.78 

1.79 

1.80 

1.82 

.84 

1.85 

1.87 

1.88 

1.90 

1.92 

1.94 

1.95 

1.97 

54 

37 

1.77 

.79 

1.80 

1.82 

1.83 

1.85 

1.86 

.88 

1.90 

1.91 

1.93 

1.95 

1.96 

1.98 

2.00 

2.02 

53 

38 

1.82 

.83 

1.84 

1.86 

1.88 

1.89 

1.91 

.92 

1.94 

1.96 

1.98 

1.99 

2.01 

2.03 

2.05 

2.07 

52 

39 

1.86 

.87 

1.89 

1.90 

1.92 

1.93 

1.95 

.97 

1.98 

2.00 

2.02 

2.04 

2.06 

2.07 

2.09 

2.11 

51 

40 

1.89 

.91 

1.93 

1.94 

1.96 

1.97 

1.99 

2.01 

2.03 

2.04 

2.06 

2.08 

2.10 

2.12 

2.14 

2.16 

50 

41 

1.93 

1.95 

1.96 

1.98 

2.00 

2.01 

2.03 

2.05 

2.07 

2.09 

2.10 

2.12 

2.14 

2.16 

2.18 

2.20 

49 

42 

1  97 

1  99 

2.00 

2.02 

2  04 

2.05 

2.07 

2.09 

2.11 

2.13 

2.15 

2.16 

2.18 

2.20 

2.22 

2.25 

48 

43 

2.01 

2.03 

2.04 

2.06 

2.08 

2.09 

2.11 

2.13 

2.15 

2.17 

2.19 

2.21 

2.23 

2.25 

2.27 

2.29 

47 

44 

2.05 

2.06 

2.08 

2.10 

2.12 

2.13 

2.1.5 

2.17 

2.19 

2.21 

2.23 

2.25 

2.27 

2.29 

2.31 

2.33 

46 

45 

2.08 

2.10 

2.12 

2.14 

2.15 

2.17 

2.19 

2.21 

2.23 

2.25 

2.27 

2.29 

2.31 

2.33 

2.35 

2.37 

45 

70°10' 

70°  20' 

70°  30' 

70°  40' 

70°  50' 

71° 

71°  Itr 

•71-W 

71*30' 

71°  40' 

71°  50' 

72° 

72°  10' 

72°  20' 

72°  30' 

72°  40' 

DETERMINATION   OF    TIME. 


71 


Table  of  factors  for  reduction  of  transit  observations, 

TOP  ARGUMENT- STAR'S  DECLINATION  (J). 

SIDE  ARGUMENT-STAR'S  ZENITH  DISTANCE  (C). 

[For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  this  paee.] 


C 

70°  10' 

70°  20' 

70°  30' 

70°  40' 

70°  50' 

71° 

71°  10' 

71°  20' 

71°  30' 

71°  40' 

71°  50' 

72° 

72°  10' 

72°  20' 

72°  30' 

72°  40' 

C 

46 

2.12 

2.14 

2.15 

2.17 

2.19 

2.21 

2.23 

2.25 

2.27 

2.29 

2.31 

2.33 

2.35 

2.37 

2.39 

2.41 

o 
44 

47 

2.16 

2.17 

2.19 

2.21 

2.23 

2.25 

2.27 

2.28 

2.30 

2.32 

2.35 

2.37 

2.39 

2.41 

2.43 

2.45 

43 

48 

2.19 

2.21 

2.22 

2.24 

2.26 

2.28 

2.30 

2.32 

2.34 

2.36 

2.38 

2.40 

2.43 

2.45 

2.47 

2.49 

42 

49 

2.22 

2.24 

2.26 

2.28 

2.30 

2.32 

2.34 

2.36 

2.38 

2.40 

2.42 

2.44 

2.46 

2.49 

2.51 

2.53 

41 

50 

2.26 

2.28 

2.29 

2.31 

2.33 

2.35 

2.37 

2.39 

2.41 

2.44 

2.46 

2.48 

2.50 

2.52 

2.55 

2.57 

40 

51 

2.29 

2.31 

2.33 

2.35 

2.37 

2.39 

2.41 

2.43 

2.45 

2.47 

2.49 

2.51 

2.54 

2.56 

2.58 

2.61 

39 

52 

2.32 

2.34 

2.36 

2.38 

2.40 

2.42 

2.44 

2.46 

2.48 

2.50 

2.53 

2.55 

2.57 

2.60 

2.62 

2.64 

38 

53 

2.35 

2.37 

2.39 

2.41 

2.43 

2.45 

2.47 

2.50 

2.52 

2.54 

2.56 

2.58 

2.61 

2.63 

2.66 

2.68 

37 

54 

2.38 

2.40 

2.42 

2.44 

2.46 

2.48 

2.51 

2.53 

2.55 

2.57 

2.60 

2.62 

2.64 

2.67 

2.69 

2.72 

36 

55 

2.41 

2.43 

2.45 

2.47 

2.50 

2.52 

2.54 

2.56 

2.58 

2.60 

2.63 

2.65 

2.68 

2.70 

2.72 

2.75 

35 

56 

2.44 

2.46 

2.48 

2.50 

2.52 

2.55 

2.57 

2.59 

2.61 

2.64 

2.66 

2.68 

2.71 

2.73 

2.76 

2.78 

34 

57 

2.47 

2.49 

2.51 

2.53 

2.55 

2.58 

2.60 

2.62 

2.64 

2.67 

2.69 

2.71 

2.74 

2.76 

2.79 

2.82 

33 

58 

2.50 

2.52 

2.54 

2.56 

2.58 

2.61 

2.63 

2.65 

2.67 

2.70 

2.72 

2.74 

2.77 

2.79 

2.82 

2.85 

32 

59 

2.53 

2.55 

2.57  i      2.59 

2.61 

2.63 

2.66 

2.68 

2.70 

2.72 

2.75 

2.77 

2.80 

2.82 

2.85 

2.88 

31 

60 

2.55 

2.57 

2.59 

2.62 

2.64 

2.66 

2.68 

2.71 

2.73 

2.75 

2.78 

2.80 

2.83 

2.85 

2.88 

2.91 

30 

61 

2.5S 

2.60 

2.62 

2.64 

2.66 

2.69 

2.71 

2.73 

2.76 

2.78 

2.80 

2.83 

2.86 

2.88 

2.91 

2.94 

29 

62 

2.60 

2.62 

2.64 

2.67 

2.69 

2.71 

2.74 

2.76 

2.78 

2.81 

2.83 

2.86 

2.88 

2.91 

2.94 

2.96 

28 

63 

2.63 

2.65 

2.67 

2.69 

2.71 

2.74 

2.76 

2.78 

2.81 

2.83 

2.86 

2.88 

2.91 

2.94 

2.96 

2.99 

27 

64 

2.65 

2.67 

2.69 

2.72 

2.74 

2.76 

2.78 

2.81 

2.83 

2.86 

2.88 

2.91 

2.94 

2.96 

2.99 

3.02 

26 

65 

2.67 

2.69 

2.71 

2.74 

2.76 

2.78 

2.81 

2.83 

2.86 

2.88 

2.91 

2.93 

2.96 

2.99 

3.01 

3.04 

25 

66 

2.69 

2.71 

2.74 

2.76 

2.78 

2.81 

2.83 

2.85 

2.88 

2.90 

2.93 

2.96 

2.98 

3.01 

3.04 

3.07 

24 

67 

2.71 

2.74 

2.76 

2.78 

2.80 

2.83 

2.85 

2.88 

2.90 

2.93 

2.95 

2.98 

3.01 

3.03 

3.06 

3.09 

23 

68 

2.73 

2.76 

2.78 

2.80 

2.82 

2.85 

2.87 

2.90 

2.92 

2.95 

2.97 

3.00 

3.03 

3.06 

3.08 

3.11 

22 

69 

2.75 

2.77 

2.80 

2.82 

2.84 

2.87 

2.89 

2.92 

2.94 

2.97 

2.99 

3.02 

3.05 

3.08 

3.10 

3.13 

21 

JO 

2.77 

2.79 

2.81 

2.84 

2.86 

2.89 

2.91 

2.94 

2.96 

2.99 

3.01 

3.04 

3.07 

3.10 

3.12 

3.15 

20 

71 

2.79 

2.81 

2.83 

2.86 

2.88 

2.90 

2.93 

2.95 

2.98 

3.01 

3.03 

3.06 

3.09 

3.12 

3.14 

3.17 

19 

72 

2.80 

2.83 

2.85 

2.87 

2.90 

2.92 

2.95 

2.97 

3.00 

3.02 

3.05 

3.08 

3.10 

3.13 

3.16 

3.19 

18 

73 

2.82 

2.84 

2.86 

2.89 

2.91 

2.94 

2.96 

2.99 

3.01 

3.04 

3.07 

3.09 

3.12 

3.15 

3.18 

3.21 

17 

74 

2.83 

2.86 

2.88 

2.90 

2.93 

2.95 

2.98 

3.00 

3.03 

3.06 

3.08 

3.11 

3.14 

3.17 

3.20 

3.23 

16 

75 

2.85 

2.87 

2.89 

2.92 

2.94 

2.97 

2.99 

3.02 

3.04 

3.07 

3.10 

3.13 

3.15 

3.18 

3.21 

3.24 

15 

76 

2.86 

2.88 

2.91 

2.93 

2.96 

2.98 

3.01 

3.03 

3.06 

3.  OS 

3.11 

3.14 

3.17 

3.20 

3.23 

3.26 

14 

77 

2.87 

2.90 

2.92 

2.94 

2.97 

2.99 

3.02 

3.04 

3.07 

3.10 

3.12 

3.15 

3.18 

3.21 

3.24 

3.27 

13 

78 

2.88 

2.91 

2.93 

2.95 

2.9S 

3.00 

3.03 

3.06 

3.  OS 

3.11 

3.14 

3.16 

3.19 

3.22 

3.25 

3.28 

12 

79 

2.89 

2.92 

2.94 

2.96 

2.99 

3.02 

3.04 

3.07 

3.09 

3.12 

3.15 

3.18 

3.20 

3.23 

3.28 

3.29 

11 

80 

2.90 

2.93 

2.95 

2.97 

3.00 

3.02 

3.05 

3.08 

3.10 

3.13 

3.16 

3.19 

3.22 

3.24 

3.27 

3.31 

10 

81 

2.91 

2.94 

2.96 

2.98 

3.01 

3.03 

3.06 

3.09 

3.11 

3.14 

3.17 

3.20 

3.23 

3.25 

3.28 

3.32 

9 

82 

2.92 

2.94 

2.97 

2.99 

3.02 

3.04 

3.07 

3.09 

3.U 

3.15 

3.18 

3.20 

3.23 

3.26 

3.29 

•    3.32 

8 

83 

2.92 

2.95 

2.97 

3.00 

3.02 

3.05 

3.08 

3.10 

3.13 

3.16 

3.18 

3.21 

3.24 

3.27 

3.30 

3.33 

7 

84 

2.93 

2.96 

2.98 

3.00 

3.03 

3.06 

3.08 

3.11 

3.13 

3.16 

3.19 

3.22 

3.25 

3.28 

3.31 

3.34 

6 

85 

2.94 

2.96 

2.98 

3.01 

3.03 

3.08 

3.09 

3.11 

3.14 

3.17 

3.20 

3.22 

3.25 

3.28 

3.31 

3.34 

5 

86 

2.94 

2.96 

2.99 

3.01 

3.04 

3.06 

3.09 

3.12 

3.14 

3.17 

3.20 

3.23 

3.26 

3.29 

3.32 

'3.35 

4 

87 

2.94 

2.97 

2.99 

3.02 

3.04 

3.07 

3.09 

3.12 

3.15 

3.18 

3.20 

3.23 

3.26 

3.29 

3.32 

3.35 

3 

88 

2.95 

2.97 

2.99 

3.02 

3.04 

3.07 

3.10 

3.12 

3.15 

3.18 

3.20 

3.23 

3.26 

3.29 

3.32 

3.35 

2 

89 

2.95 

2.97 

3.00 

3.02 

3.04 

3.07 

3.10 

3.12 

3.15 

3.18 

3.21 

3.24 

3.27 

3.30 

3.33 

3.36 

1 

90 

2.95 

2.97 

3.00 

3.02 

3.05 

3.07 

3.10 

3.12 

3.15 

3.18 

3.21 

3.24 

3.27 

3.30 

3.33 

3.36 

0 

70°  10' 

70°20' 

70°  30' 

70°  40' 

70°  50' 

71° 

71°  10' 

71°  20' 

71°  30' 

71°  40" 

71°  5V 

72° 

72°  107 

72°  20' 

72°  30' 

72°  40' 

72 


U.   S.   COAST   AND   GEODETIC   SUKVEY   SPECIAL   PUBLICATION    NO.   14. 


Table  of  factors  for  reduction  of  transit  observations. 

TOP  ARGUMENT- STAR'S  DECLINATION  (3). 
SIDE  ARGUMENT- STAR'S  ZENITH  DISTANCE  (C). 
[For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  opposite  page.] 


C 

72"  40' 

72°  50' 

73° 

73°  ICC 

73°  20' 

73°  30' 

73°  40' 

73°  50' 

74° 

74°  W 

74°  20' 

74°  30' 

74°  40' 

74°  50' 

75° 

75°  10' 

<. 

1 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.07 

.07 

.07 

.07 

89 

2 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.12 

.13 

.13 

.13 

.13 

.13 

.13 

.13 

.14 

88 

3 

.18 

.18 

.18 

.18 

.18 

.18 

.19 

.19 

.19 

.19 

.19 

.20 

.20 

.20 

.20 

.20 

87 

4 

.23 

.24 

.24 

.24 

.24 

.24 

.25 

.25 

.25 

.26 

.26 

.26 

.26 

.27 

.27 

.27 

86 

5 

.29 

.30 

.30 

.30 

.30 

.31 

.31 

.31 

.32 

.32 

.32 

.33 

.33 

.33 

.34 

.34 

85 

6 

.35 

.35 

.36 

.36 

.36 

.37 

.37 

.38 

.38 

.38 

.39 

.39 

.40 

.40 

.40 

.41 

84 

7 

.41 

.41 

.42 

.42 

.42 

.43 

.43 

.44 

.44 

.45 

.45 

.46 

.46 

.47 

.47 

.48 

83 

8 

.47 

.47 

.48 

.48 

.48 

.49 

.50 

.50 

.50 

.51 

.52 

.52 

.53 

.53 

.54 

.54 

82 

9 

.52 

.53 

.53 

.54 

.54 

.55 

.56 

.56 

.57 

.57 

.58 

.58 

.59 

.60 

.60 

.61 

81 

10 

.58 

.59 

.59 

.60 

.60 

.61 

.62 

.62 

.63 

.64 

.64 

.65 

.66 

.66 

.67 

.68 

80 

11 

.64 

.65 

.65 

.66 

.66 

.67 

.68 

.68 

.69 

.70 

.71 

.71 

.72 

.73 

.74 

.74 

79 

12 

.70 

.70 

.71 

.72 

.72 

.73 

.74 

.75 

.75 

.76 

.77 

.78 

.79 

.79 

.80 

.81 

78 

13 

.76 

.76 

.77 

.78 

.78 

.79 

.80 

.81 

.82 

.82 

.83 

.84 

.85 

.86 

.87 

.88 

77 

14 

.81 

.82 

.83 

.84 

.84 

.85 

.86 

.87 

.88 

.89 

.90 

.91 

.92 

.93 

.94 

.95 

76 

15 

.87 

.88 

.89 

.89 

.90 

.91 

.92 

.93 

.94 

.95 

.96 

.97 

.98 

.99 

1.00 

1.01 

75 

16 

.92 

.93 

.94 

.95 

.96 

.97 

.98 

.99 

1.00 

1.01 

1.02 

1.03 

1.04 

1.05 

1.06 

1.08 

74 

17 

.98 

.99 

1.00 

1.01 

1.02 

1.03 

1.04 

1.05 

1.06 

1.07 

1.08 

1.09 

1.11 

1.12 

1.13 

1.14 

73 

18 

1.04 

.05 

1.06 

1.07 

1.08 

1.09 

1.10 

1.11 

1.12 

1.13 

1.14 

1.16 

1.17 

1.18 

1.19 

1.21 

72 

19 

1.09 

.10 

1.11 

1.12 

1.14 

1.15 

1.16 

1.17 

1.18 

1.19 

1.21 

1.22 

1.23 

1.24 

1.26 

1.27 

71 

20 

1.15 

.16 

1.17 

1.18 

1.19 

1.20 

1.22 

1.23 

1.24 

1.25 

1.27 

1.28 

1.29 

1.31 

1.32 

1.34 

70 

21 

1.20 

.21 

1.22 

1.24 

1.25 

1.26 

.27 

1.29 

1.30 

1.31 

1.33 

1.34 

1.36 

1.37 

1.38 

1.40 

69 

22 

1.26 

.27 

1.28 

1.29 

1.31 

1.32 

.33 

1.34 

1.36 

1.37 

1.39 

1.40 

1.42 

1.43 

1.45 

1.46 

68 

23 

1.31 

.32 

1.34 

1.35 

1.36 

1.38 

.39 

1.40 

1.42 

1.43 

1.45 

1.46 

1.48 

1.49 

1.51 

1.53 

67 

24 

1.36 

.38 

.39 

1.40 

1.42 

1.43 

.45 

1.46 

1.48 

1.49 

1.51 

1.52 

1.54 

1.55 

1.57 

1.59 

66 

25 

1.42 

.43 

.45 

1.46 

1.47 

1.49 

.50 

1.52 

1.53 

1.55 

1.56 

1.58 

1.60 

1.62 

1.63 

1.65 

65 

26 

1.47 

1.48 

.50 

1.51 

1.53 

1.54 

1.56 

1.58 

1.59 

1.61 

1.62 

1.64 

1.66 

1.68 

1.69 

1.71 

64 

27 

1.52 

1.54 

.55 

1.57 

1.58 

1.60 

1.61 

1.63 

1.65 

1.66 

1.68 

1.70 

1.72 

1.74 

1.75 

1.77 

63 

28 

1.58 

1.59 

.60 

1.62 

1.64 

1.65 

1.67 

1.69 

1.70 

1.72 

1.74 

1.76 

1.78 

1.79 

1.81 

1.83 

62 

29 

1.63 

1.64 

.66 

1.67 

1.69 

1.71 

1.72 

1.74 

1.76 

1.78 

1.80 

1.81 

1.83 

1.85 

1.87 

1.89 

61 

30 

1.68 

1.69 

.71 

1.73 

1.74 

1.76 

1.78 

1.80 

1.81 

1.83 

1.85 

1.87 

1.89 

1.91 

1.93 

1.95 

60 

31 

1.73 

1.74 

.76 

1.78 

1.80 

1.81 

1.83 

1.85 

1.87 

1.89 

1.91 

1.93 

1.95 

1.97 

1.99 

2.01 

59 

32 

1.78 

1.80 

.81 

1.83 

1.85 

1.87 

1.88 

1.90 

1.92 

1.94 

1.96 

1.98 

2.00 

2.02 

2.05 

2.07 

58 

33 

1.83 

1.85 

.86 

1.88 

1.90 

1.92 

1.94 

1.96 

1.98 

2.00 

2.02 

2.04 

2.06 

2.08 

2.10 

2.13 

57 

34 

1.88 

1.89 

.91 

1.93 

1.95 

1.97 

1.99 

2.01 

2.03 

2.05 

2.07 

2.09 

2.12 

2.14 

2.16 

2.18 

56 

35 

1.92 

1.94 

.96 

1.98 

2.00 

2.02 

2.04 

2.06 

2.08 

2.10 

2.12 

2.15 

2.17 

2.19 

2.22 

2.24 

55 

36 

1.97 

1.99 

2.01 

2.03 

2.05 

2.07 

2.09 

2.11 

2.13 

2.15 

2.18 

2.20 

2.22 

2.25 

2.27 

2.30 

54 

37 

2.02 

2.04 

2.06 

2.08 

2.10 

2.12 

2.14 

2.16 

2.18 

2.21 

2.23 

2.25 

2.28 

2.30 

2.33 

2.35 

53 

38 

2.07 

2.09 

2.11 

2.13 

2.15 

2.17 

2.19 

2.21 

2.23 

2.26 

2.28 

2.30 

2.33 

2.35 

2.38 

2.40 

52 

39 

2.11 

2.13 

2.15 

2.17 

2.19 

2.22 

2.24 

2.26 

2.28 

2.31 

2.33 

2.35 

2.38 

2.40 

2.43 

2.46 

51 

40 

2.16 

2.18 

2.20 

2.22 

2.24 

2.26 

2.29 

2.31 

2.33 

2.36 

2.38 

2.40 

2.43 

2.46 

2.48 

2.51 

50 

41 

2.20 

2.22 

2.24 

2.26 

2.29 

2.31 

2.33 

2.36 

2.38 

2.40 

2.43 

2.45 

2.48 

2.51 

2.53 

2.56 

49 

42 

2.25 

2.27 

2.29 

2.31 

2.33 

2.36 

2.38 

2.40 

2.43 

2.45 

2.48 

2.50 

2.53 

2.56 

2.58 

2.61 

48 

43 

2.20 

2.31 

2.33 

2.36 

2.38 

2.40 

2.42 

2.45 

2.47 

2.50 

2.53 

2.55 

2.58 

2.61 

2.63 

2.66 

47 

44 

2.33 

2.35 

2.38 

2.40 

2.42 

2.45 

2.47 

2.50 

2.52 

2.55 

2.57 

2.60 

2.63 

2.66 

2.68 

2.71 

46 

45 

2.37 

2.40 

2.42 

2.44 

2.46 

2.49 

2.51 

2.54 

2.56 

2.59 

2.62 

2.65 

2.67 

2.70 

2.73 

2.76 

45 

72°  40' 

72°  50' 

73° 

73°  10' 

73°  20' 

73°  30' 

73°  40' 

73°  50' 

74" 

74°  10' 

74°  20' 

74°  30' 

74°  40' 

74°  50' 

75° 

75°  HC 

DETERMINATION   OF   TIME. 


73 


Table  of  factors  for  reduction  of  transit  observations. 

TOP  ARGUMENT- STAR'S  DECLINATION  (3). 

SIDE   ARGUMENT- STAR'S  ZENITH  DISTANCE  (C). 

[For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  this  page.] 


C 

72"  40' 

72°  50' 

73° 

73°  10' 

73°20' 

73°  30' 

73-  40' 

73-50' 

74° 

74°  10' 

74-20' 

74-30' 

74-40' 

74-50' 

75° 

75-10' 

C 

46 

2.41 

2.44 

2.46 

2.48 

2.51 

2.53 

2.56 

2.58 

2.61 

2.64 

2.66 

2.69 

2.72 

2.75 

2.78 

2.81 

44 

47 

2.45 

2.48 

2.50 

2.52 

2.55 

2.57 

2.60 

2.63 

2.65 

2.6'i 

2.71 

2.74 

2.77 

2.80 

2.83 

2.86 

43 

48 

2.49 

2.52 

2.54 

2.57 

2.59 

2.62 

2.64 

2.67 

2.70 

2.72 

2.75 

2.78 

2.81 

2.84 

2.87 

2.90 

42 

49 

2.53 

2.56 

2.58 

2.61 

2.63 

2.66 

2.68 

2.71 

2.74 

2.77 

2.80 

2.82 

2.85 

2.88 

2.92 

2.95 

41 

50 

2.57 

2.60 

2.62 

2.64 

2.67 

2.70 

2.72 

2.75 

2.78 

2.81 

2.84 

2.87 

2.90 

2.93 

2.96 

2.99 

40 

51 

2.61 

2.63 

2.66 

2.6S 

2.71 

2.74 

2.76 

2.79 

2.82 

2.85 

2.88 

2.91 

2.94 

2.97 

3.00 

3.04 

39 

52 

2.64 

2.67 

2.69 

2.72 

2.75 

2.77 

2.80 

2.83 

2.86 

2.89 

2.92 

2.95 

2.98 

3.01 

3.04 

3.08 

33 

53 

2.68 

2.71 

2.73 

2.76 

2.78 

2.81 

2.84 

2.87 

2.90 

2.93 

2.96 

2.99 

3.02 

3.05 

3.09 

3.12 

37 

54 

2.72 

2.74 

2.77 

2.79 

2.82 

2.85 

2.88 

2.91 

2.94 

2.97 

3.00 

3.03 

3.06 

3.09 

3.13 

3.16 

36 

55 

2.75 

2.78 

2.80 

2.83 

2.86 

2.88 

2.91 

2.94 

2.97 

3.00 

3.03 

3.07 

3.10 

3.13 

3.16 

3.20 

35 

56 

2.78 

2.81 

2.84 

2.86 

2.89 

2.92 

2.95 

2.98 

3.01 

3.04 

3.07 

3.10 

3.14 

3.17 

3.20 

3.24 

'     34 

57 

2.82 

2.84 

2.87 

2.90 

2.92 

2.93 

2.98 

3.01 

3.04 

3.07 

3.11 

3.14 

3.17 

3.21 

3.24 

3.28 

33 

58 

2.85 

2.87 

2.90 

2.93 

2.96 

2.99 

3.02 

3.05 

3.08 

3.11 

3.14 

3.17 

3.21 

3.24 

3.23 

3.31 

32 

59 

2.88 

2.90 

2.93 

2.% 

2.99 

3.02 

3.05 

3.08 

3.11 

3.14 

3.17 

3.21 

3.24 

3.28 

3.31 

3.35 

31 

60 

2.91 

2.93 

2.96 

2.99 

3.02 

3.05 

3.08 

3.11 

3.14 

3.17 

3.21 

3.24 

3.28 

3.31 

3.35 

3.3S 

30 

61 

2.94 

2.96 

2.99 

3.02 

3.05 

3.08 

3.11 

3.14 

3.17 

3.21 

3.24 

3.27 

3.31 

3.34 

3.38 

3.42 

29 

62 

2.96 

2.99 

3.02 

3.05 

3.08 

3.11 

3.14 

3.17 

3.20 

3.24 

3.27 

3.30 

3.34 

3.3S 

3.41 

3.45 

28 

63 

2.99 

3.02 

3.05 

3.  OS 

3.11 

3.14 

3.17 

3.20 

3.23 

3.27 

3.30 

3.33 

3.37 

3.41 

3.44 

3.43 

27 

64 

3.02 

3.04 

3.07 

3.10 

3.13 

3.16 

3.20 

3.23 

3.26 

3.29 

3.33 

3.36 

3.40 

3.44 

3.47 

3.51 

26 

65 

3.04 

3.07 

3.10 

3.13 

3.16 

3.19 

3.22 

3.26 

3.29 

3.32 

3.36 

3.39 

3.43 

3.46 

3.50 

3.54 

25 

66 

3.07 

3.10 

3.13 

3.16 

3.18 

3.22 

3.25 

3.28 

3.31 

3.35 

3.38 

3.42 

3.46 

3.49 

3.53 

3.57 

24 

67 

3.09 

3.12 

3.15 

3.18 

3.21 

3.24 

3.27 

3.31 

3.34 

3.37 

3.41 

3.44 

3.48 

3.52 

3.56 

3.60 

23 

68 

3.11 

3.14 

3.17 

3.20 

3.23 

3.26 

3.30 

3.33 

3.36 

3.40 

3.43 

3.47 

3.51 

3.54 

3.58 

3.62 

22 

69 

3.13 

3.16 

3.19 

3.22 

3.26 

3.29 

3.32 

3.35 

3.39 

3.42 

3.46 

3.49 

3.53 

3.57 

3.61 

3.65 

21 

70 

3.15 

3.18 

3.21 

3.24 

3.28 

3.31 

3.34 

3.38 

3.41 

3.44 

3.48 

3.52 

3.55 

3.59 

3.63 

3.67 

20 

71 

3.17 

3.20 

3.23 

3.26 

3.30 

3.33 

3.36 

3.40 

3.43 

3.47 

3.50 

3.54 

3.58 

3.61 

3.65 

3.69 

19 

72 

3.19 

3.22 

3.25 

3.28 

3.32 

3.35 

3.38 

3.42 

3.45 

3.49 

3.52 

3.56 

3.60 

3.63 

3.67 

3.72 

IS 

73 

3.21 

3.24 

3.27 

3.30 

?.33 

3.37 

3.40 

3.44 

3.47 

3.50 

3.54 

3.58 

3.62 

3.65 

3.69 

3.74 

17 

74 

3.23 

3.26 

3.29 

3.32 

3.35 

3.38 

3.42 

3.45 

3.49 

3.52 

3.56 

3.60 

3.64 

3.67 

3.71 

3.76 

16 

75 

3.24 

3.27 

3.30 

3.34 

3.37 

3.40 

3.44 

3.47 

3.5C 

3.54 

3.58 

3.61 

3.65 

3.69 

3.73 

3.77 

15 

76 

3.26 

3.29 

3.32 

3.35 

3.38 

3.42 

3.45 

3.48 

3.52 

3.56 

3.59 

3.63 

3.67 

3.71 

3.75 

3.79 

14 

77 

3.27 

3.30 

3.33 

3.36 

3.40 

3.43 

3.46 

3.50 

3.54 

3.57 

3.61 

3.65 

3.68 

3.72 

3.76 

3.81 

13 

78 

3.28 

3.31 

3.34 

3.38 

3.41 

3.44 

3.48 

3.51 

3.55 

3.58 

3.62 

3.66 

3.70 

3.74 

3.78 

3.82 

12 

79 

3.29 

3.33 

3.36 

3.39 

3.42 

3.46 

3.49 

3.53 

3.56 

3.60 

3.64 

3.67 

3.71 

3.75 

3.79 

3.83 

11 

80 

3.31 

3.34 

3.37 

3.40 

3.43 

3.47 

3.50 

3.54 

3.57 

3.61 

3.65 

3.68 

3.72 

3.76 

3.81 

3.85 

10 

81 

3.32 

3.35 

3.38 

3.41 

3.44 

3.48 

3.51 

3.55 

3.58 

3.62 

3.66 

3.70 

3.74 

3.78 

3.82 

3.86 

9 

82 

3.32 

3.36 

3.39 

3.42 

3.45 

3.49 

3.52 

3.56 

3.59 

3.63 

3.67 

3.71 

3.75 

3.79 

3.83 

3.87 

8 

83 

3.33 

3.36 

3.40 

3.43 

3.46 

3.49 

3.53 

3.56 

3.60 

3.64 

3.68 

3.72 

3.75 

3.79 

3.84 

3.88 

7 

84 

3.34 

3.37 

3.40 

3.43 

3.47 

3.50 

3.54 

3.57 

3.61 

3.64 

3.68 

3.72 

3.76 

3.80 

3.84 

3.88 

6 

85 

3.34 

3.38 

3.41 

3.44 

3.47 

3.51 

3.54 

3.58 

3.61 

3.65 

3.69 

3.73 

3.77 

3.81 

3.85 

3.89 

5 

86 

3.35 

3.38 

3.41 

3.44 

3.48 

3.51 

3.55 

3.58 

3.62 

3.66 

3.69 

3.73 

3.77 

3.81 

3.85 

3.90 

4 

87 

3.35 

3.38 

3.42 

3.45 

3.48 

3.52 

3.55 

3.59 

3.62 

3.66 

3.70 

3.74 

3.78 

3.82 

3.86 

3.90 

3 

88 

3.35 

3.39 

3.42 

3.45 

3.48 

3.52 

3.55 

3.59 

3.62 

3.66 

3.70 

3.74 

3.78 

3.82 

3.86 

3.90 

2 

89 

3.36 

3.39 

3.42 

3.45 

3.49 

3.52 

3.56 

3.59 

3.63 

3.66 

3.70 

3.74 

3.78 

3.82 

3.86 

3.91 

1 

90 

3.36 

3.39 

3.42 

3.45 

3.49 

3.52 

3.56 

3.59 

3.63 

3.66 

3.70 

3.74 

3.78 

3.82 

3.86 

3.91 

0 

72-40' 

72°  50' 

73° 

73°  10* 

73°  20' 

73°  30! 

73°  40' 

73-50' 

74° 

74°10/ 

74°  20' 

74-30' 

74°  W 

74-50' 

75° 

75°  W 

74 


U.   S.   COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 


Table  of  factors  for  reduction  of  transit  observations. 

TOP  ARGUMENT- STAR'S  DECLINATION  («). 

SIDE   ARGUMENT- STAR'S  ZENITH  DISTANCE  (C). 

[For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  opposite  page.] 


C 

75°  W 

75°  20' 

75°  30' 

75°  40' 

75°  50' 

J6° 

76°  HK 

76°  20' 

76°  30' 

76°  40' 

76-50' 

77° 

77°  10' 

77°  20' 

77°  30' 

77°  40' 

C 

1 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

89 

2 

.14 

.14 

.14 

.14 

.14 

.14 

.15 

.15 

.15 

.15 

.15 

.16 

.16 

.16 

.16 

.16 

88 

3 

.20 

.21 

.21 

.21 

.21 

.22 

.22 

.22 

.22 

.23 

.23 

.23 

.24 

.24 

.24 

.24 

87 

4 

.27 

.28 

.28 

.28 

.28 

.29 

.29 

.30 

.30 

.30 

.31 

.31 

.31 

.32 

.32 

.33 

86 

5 

.34 

.34 

.35 

.35 

.36 

.36 

.36 

.37 

.37 

.38 

.38 

.39 

.39 

.40 

.40 

.41 

85 

6 

.41 

.41 

.42 

.42 

.43 

.43 

.44 

.44 

.45 

.45 

.46 

.46 

.47 

.48 

.48 

.49 

84 

7 

.48 

.48 

.49 

.49 

.50 

.50 

.51 

.52 

.52 

.53 

.54 

.54 

.55 

.56 

.56 

.57 

83 

g 

.54 

.55 

.56 

.56 

.57 

.58 

.58 

.59 

.60 

.60 

.61 

.62 

.63 

.64 

.64 

.65 

82 

9 

.61 

.62 

.62 

.63 

.64 

.65 

.65 

.66 

.67 

.68 

.69 

.70 

.70 

.71 

.72 

.73 

81 

10 

.68 

.69 

.69 

.70 

.71 

.72 

.73 

.74 

.74 

.75 

.76 

.  77 

.78 

.79 

.80 

.81 

80 

11 

1      .74 

.75 

.76 

.77 

.78 

.79 

.80 

.81 

.82 

.83 

.84 

.85 

.86 

.87 

.88 

.89 

79 

12 

.81 

.82 

.83 

.84 

.85 

.86 

.87 

.88 

.89 

.90 

*.91 

.92 

.94 

.95 

.96 

.97 

78 

13 

.88 

.89 

.90 

.91 

.92 

.93 

.94 

.95 

.96 

.98 

.99 

1.00 

1.01 

1.03 

1.04 

1.05 

77 

14 

.95 

.96 

.97 

.98 

.99 

1.00 

1.01 

1.02 

.04 

1.05 

1.06 

1.08 

1.09 

1.10 

1.12 

1.13 

76 

15 

1.01 

1.02 

1.03 

1.04 

1.08 

1.07 

1.08 

1.10 

.11 

1.12 

1.14 

1.15 

1.16 

1.18 

1.20 

1.21 

75 

16 

.08 

1.09 

1.10 

1.11 

1.13 

1.14 

1.15 

1.17 

.18 

1.20 

1.21 

1.23 

1.24 

1.26 

1.28 

1.29 

74 

17 

.14 

1.16 

1.17 

1.18 

1.20 

1.21 

1.22 

1.24 

.25 

1.27 

1.28 

1.30 

1.32 

1.33 

1.35 

1.37 

73 

18 

.21 

1.22 

1.23 

1.25 

1.26 

1.28 

1.29 

1.31 

.32 

1.34 

1.36 

1.37 

1.39 

1.41 

1.43 

1.45 

72 

19 

.27 

1.29 

1.30 

1.32 

1.33 

1.35 

1.36 

1.38 

.39 

1.41 

1.43 

1.45 

1.47 

1.48 

1.50 

1.52 

71 

•20 

.34 

1.35 

1.37 

1.38 

1.40 

1.41 

1.43 

1.45 

.47 

1.48 

1.50 

1.52 

1.54 

1.56 

1.58 

1.60 

70 

21 

.40 

1.12 

1.43 

1.45 

1.46 

1.48 

.50 

1.52 

.54 

1.55 

1.57 

1.59 

1.61 

1.63 

1.65 

1.68 

69 

22 

.46 

1.48 

1.50 

1.51 

1.53 

1.55 

.57 

1.58 

.60 

1.62 

1.64 

1.66 

1.69 

1.71 

1.73 

1.75 

68 

23 

.53 

1.54 

1.56 

1.58 

1.60 

1.62 

.63 

1.65 

.67 

1.69 

1.72 

1.74 

1.76 

1.78 

1.81 

1.83 

67 

24 

.59 

1.61 

1.63 

1.64 

1.66 

1.68 

.70 

1.72 

.74 

1.76 

1.79 

1.81 

1.83 

1.86 

1.88 

1.90 

66 

25 

.65 

1.67 

1.69 

1.71 

1.73 

1.75 

.77 

1.79 

.81 

1.83 

1.86 

1.88 

1.90 

1.93 

1.95 

1.98 

65 

26 

.71 

1.73 

1.75 

1.77 

1.79 

1.81 

.83 

1.86 

.88 

1.90 

1.92 

1.95 

1.97 

2.00 

2.02 

2.05 

64 

27 

.77 

1.79 

1.81 

1.83 

1.86 

1.88 

.90 

1.92 

.95 

1.97 

1.99 

2.02 

2.04 

2.07 

2.10 

2.12 

63 

23 

.83 

1.85 

1.87 

1.90 

1.92 

1.94 

.96 

1.99 

2.01 

2.04 

2.06 

2.09 

2.11 

2.14 

2.17 

2.20 

62 

29 

1.89 

1.92 

1.94 

1.96 

1.98 

2.00 

2.03 

2.05 

2.08 

2.10 

2.13 

2.15 

2.18 

2.21 

2.24 

2.27 

61 

30 

1.95 

1.98 

2.00 

2.02 

2.04 

2.07 

2.09 

2.12 

2.14 

2.17 

2.20 

2.22 

2.25 

2.28 

2.31 

2.34 

60 

31 

2.01 

2.03 

2.06 

2.08 

2.10 

2.13 

2.15 

2.18 

2.21 

2.23 

2.26 

2.29 

2.32 

2.35 

2.38 

2.41 

59 

32 

2.07 

2.09 

2.12 

2.14 

2.16 

2.19 

2.22 

2.24 

2.27 

2.30 

2.33 

2.36 

2.39 

2.42 

2.45 

2.48 

58 

33 

2.13 

2.15 

2.18 

2.28 

2.22 

2.25 

2.28 

2.30 

2.33 

2.36 

2.39 

2.42 

2.45 

2.48 

2.52 

2.55 

57 

34 

2.18 

2.21 

2.23 

2.26 

2.28 

2.31 

2.34 

2.37 

2.40 

2.42 

2.46 

2.49 

2.52 

2.55 

2.58 

2.62 

56 

35 

2.24 

2.26 

2.29 

2.32 

2.34 

2.37 

2.40 

2.43 

2.46 

2.49 

2.52 

2.55 

2.58 

2.62 

2.65 

2.68 

55 

36 

2.30 

2.32 

2.35 

2.37 

2.40 

2.43 

2.46 

2.49 

2.52 

2.55 

2.58 

2.61 

2.65 

2.68 

2.72 

2.75 

54 

37 

2.35 

2.38 

2.40 

2.43 

2.46 

2.49 

2.52 

2.55 

2.58 

2.61 

2.64 

2.67 

2.71 

2.74 

2.78 

2.82 

53 

38 

2.40 

2.43 

2.46 

2.49 

2.52 

2.55 

2.58 

2.61 

2.64 

2.67 

2.70 

2.74 

2.77 

2.81 

2.85 

2.88 

52 

39 

2.46 

2.49 

2.51 

2.54 

2.57 

2.60 

2.63 

2.66 

2.70 

2.73 

2.76 

2.80 

2.83 

2.87 

2.91 

2.95 

51 

40 

2.51 

2.54 

2.57 

2.60 

2.63 

2.66 

2.69 

2.72 

2.75 

2.79 

2.82 

2.86 

2.89 

2.93 

2.97 

3.01 

50 

41 

2.56 

2.59 

2.62 

2.65 

2.68 

2.71 

2.74 

2.78 

2.81 

2.84 

2.88 

2.92 

2.95 

2.99 

3.03 

3.07 

49 

42 

2.61 

2.64 

2.67 

2.70 

2.73 

2.77 

2.80 

2.83 

2.87 

2.90 

2.94 

2.97 

3.01 

3.05 

3.09 

3.13 

48 

43 

2.66 

2.69 

2.72 

2.76 

2.79 

2.82 

2.85 

2.89 

2.92 

2.96 

2.99 

3.03 

3.07 

3.11 

3.15 

3.19 

47 

44 

2.71 

2.74 

2.77 

2.81 

2.84 

2.87 

2.90 

2.94 

2.98 

3.01 

3.05 

3.09 

3.13 

3.17 

3.21 

3.25 

46 

45 

2.76 

2.79 

2.82 

2.86 

2.89 

2.92 

2.96 

2.99 

3.03 

3.07 

3.10 

3.14 

3.18 

3.22 

3.27 

3.31 

45 

75°  10' 

75°  20' 

75°  30' 

75°  40' 

75°  50' 

76° 

76°  10' 

76-20' 

76°  30' 

76°  40' 

76°  50' 

77° 

77°  10' 

77°  20' 

77°  30' 

77"  W 

DETEEMINATION    OF    TIME. 


75 


Table  of  factors  for  reduction  of  transit  observations. 

TOP  ARGUMENT- STAR'S  DECLINATION  («). 

SIDE  ARGUMENT- STAR'S  ZENITH  DISTANCE  (C). 

[For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  this  page.] 


C 

75°  10' 

75°  20' 

75°  30' 

75"  40' 

75°  50' 

78° 

76°  IV 

76°  20' 

76°  30' 

76°  40' 

76°  50' 

77° 

77°  10' 

77°  20' 

77°  30' 

77°  40' 

C 

46 

2.81 

2.84 

2.87 

2.91 

2.94 

2.97 

3.01 

3.04 

3.08 

3.12 

3.16 

3.20 

3.24 

3.28 

3.32 

3.37 

44 

47 

2.86 

2.89 

2.92 

2.95 

2.99 

3.02 

3.06 

3.10 

3.13 

3.17 

3.21 

3.25 

3.29 

3.34 

3.38 

3.42 

43 

48 

2.90 

2.94 

2.97 

3.00 

3.04 

3.07 

3.11 

3.15 

3.18 

3.22 

3.26 

3.30 

3.35 

3.39 

3.43 

3.48 

42 

49 

2.95 

2.98 

3.01 

3.05 

3.08 

3.12 

3.16 

3.19 

3.23 

3.27 

3.31 

3.36 

3.40 

3.44 

3.49 

3.53 

41 

50 

2.99 

3.02 

3.06 

3.09 

3.13 

3.17 

3.20 

3.24 

3.28 

3.32 

3.36 

3.41 

3.45 

3.49 

3.54 

3.59 

40 

51 

3.04 

3.07 

3.10 

3.14 

3.18 

3.21 

3.25 

3.29 

3.33 

3.37 

3.41 

3.45 

3.50 

3.54 

3.59 

3.64 

39 

52 

3.08 

3.11 

3.15 

3.18 

3.22 

3.26 

3.30 

3.34 

3.38 

3.42 

3.46 

3.50 

3.55 

3.59 

3.64 

3.69 

38 

53 

3.12 

3.15 

3.19 

?.23 

3.26 

3.30 

3.34 

3.38 

3.42 

3.46 

3.51 

3.55 

3.60 

3.64 

3.69 

3.74 

37 

54 

3.16 

3.20 

3.23 

3.27 

3.31 

3.34 

3.38 

3.42 

3.47 

3.51 

3.55 

3.60 

3.64 

3.69 

3.74 

3.79 

36 

55 

3.20 

3.24 

3.27 

3.31 

3.35 

3.39 

3.43 

3.47 

3.51 

3.55 

3.60 

3.64 

3.69 

3.74 

3.78 

3.83 

35 

56 

3.24 

3.27 

3.31 

3.35 

3.39 

3.43 

3.47 

3.51 

3.55 

3.60 

3.64 

3.68 

3.73 

3.78 

3.83 

3.88 

34 

57 

3.28 

3.31 

3.35 

3.39 

3.43 

3.47 

3.51 

3.55 

3.59 

3.64 

3.6S 

3.73 

3.78 

3.83 

3.88 

3.93 

33 

58 

3.31 

3.35 

3.39 

3.43 

3.47 

3.51 

3.55 

3.59 

3.63 

3.68 

3.72 

3.77 

3.82 

3.87 

3.92 

3.97 

32 

59 

3.35 

3.38 

3.42 

3.46 

3.50 

3.54 

3.58 

3.63 

3.67 

3.72 

3.76 

3.81 

3.86 

3.91 

3.96 

4.01 

31 

60 

3.38 

3.42 

3.46 

3.50 

3.54 

3.58 

3.62 

3.66 

3.71 

3.76 

3.80 

3.85 

3.90 

3.95 

4.00 

4.05 

30 

61 

3.42 

3.45 

3.49 

3.53 

3.57 

3.62 

3.66 

3.70 

3.75 

3.79 

3.84 

3.89 

3.94 

3.99 

.04 

4.09 

29 

62 

3.45 

3.49 

3.53 

3.57 

3.61 

3.65 

3.69 

3.74 

3.78 

3.83 

3.88 

3.93 

3.98 

4.03 

.08 

4.13 

28 

63 

3.48 

3.52 

3.56 

3.60 

3.64 

3.68 

3.73 

3.77 

3.82 

3.86 

3.91 

3.96 

4.01 

4.06 

.12 

4.17 

27 

64 

3.51 

3.55 

3.59 

3.63 

3.67 

3.72 

3.76 

3.80 

3.85 

3.90 

3.95 

4.00 

4.05 

4.10 

.15 

4.21 

26 

65 

3.54 

3.58 

3.62 

3.66 

3.70 

3.75 

3.79 

3.84 

3.88 

3.93 

3.98 

4.03 

4.08 

4.13 

.19 

4.24 

25 

66 

3.57 

3.61 

3.65 

3.69 

3.73 

3.78 

3.82 

3.87 

3.91 

3.96 

4.01 

4.06 

4.11 

4.17 

.22 

4.28 

24 

67 

3.60 

3.64 

3.68 

3.72 

3.76 

3.81 

3.  &5 

3.90 

3.94 

3.99 

4.04 

4.09 

4.14 

4.20 

.25 

4.31 

23 

68 

3.62 

3.66 

3.70 

3.74 

3.79 

3.83 

3.88 

3.92 

3.97 

4.02 

4.07 

4.12 

4.17 

4.23 

.28 

4.34 

22 

69 

3.65 

3.69 

3.73 

3.77 

3.82 

3.86 

3.90 

3.95 

.00 

4.05 

4.10 

4.15 

4.20 

4.26 

.31 

4.37 

21 

70 

3.67 

3.71 

3.75 

3.80 

3.84 

3.89 

3.93 

3.98 

.03 

4.08 

4.12 

4.18 

4.23 

4.28 

.34 

4.40 

20 

71 

3.69 

3.73 

3.78 

3.82 

3.86 

3.91 

3.96 

.00 

.05 

4.10 

4.15 

4.20 

4.26 

4.31 

.37 

4.43 

19 

72 

3.72 

3.76 

3.80 

3.84 

3.89 

3.93 

3.98 

.02 

.07 

4.12 

4.18 

4.23 

4.28 

4.34 

.39 

4.45 

18 

73 

3.74 

3.78 

3.82 

3.86 

3.91 

3.95 

4.00 

.05 

.10 

4.15 

4.20 

4.25 

4.30 

4.36 

.42 

4.48 

17 

74 

3.76 

3.80 

3.84 

3.88 

3.93 

3.97 

4.02 

.07 

.12 

4.17 

4.22 

4.27 

4.33 

4.38 

.44 

4.50 

16 

75 

3.77 

0  DO 
O.  O4 

3.86 

3.90 

3.95 

3.99 

4.04 

.09 

.14 

4.19 

4.24 

4.29 

4.35 

4.40 

.46 

4.52 

15 

76 

3.79 

3.83 

3.88 

3.92 

3.96 

4.01 

4.06 

.11 

.16 

4.21 

4.26 

4.31 

4.37 

4.42 

.48 

4.54 

14 

77 

3.81 

3.85 

3.89 

3.94 

3.98 

4.03 

4.08 

.12 

.17 

4.22 

4.28 

4.33 

4.39 

4.44 

.50 

4.56 

13 

78 

3.82 

3.86 

3.91 

3.95 

4.00 

4.04 

4.09 

.14 

.19 

4.24 

4.29 

4.35 

4.40 

4.46 

.52 

4.58 

12 

79 

3.83 

3.88 

3.92 

3.96 

4.01 

4.06 

4.11 

4.16 

4.21 

4.26 

4.31 

4.36 

4.42 

4.48 

.54 

4.60 

11 

80 

3.85 

3.89 

3.93 

3.98 

4.02 

4.07 

4.12 

4.17 

4.22 

4.27 

4.32 

4.38 

4.43 

4.49 

.55 

4.61 

10 

81 

3.86 

3.90 

3.94 

3.99 

4.04 

4.08 

4.13 

4.18 

4.23 

4.28 

4.34 

4.39 

.45 

4.50 

.56 

4.62 

9 

82 

3.87 

3.91 

3.96 

4.00 

4.05 

4.09 

4.14 

4.19 

4.24 

4.29 

4.35 

4.40 

.46 

4.52 

.58 

4.64 

8 

83 

3.88 

3.92 

3.96 

4.01 

4.06 

4.10 

4.15 

4.20 

4.25 

4.30 

4.36 

4.41 

.47 

4.53 

.59 

4.65 

7 

84 

3.88 

3.93 

3.97 

4.02 

4.06 

4.11 

4.16 

4.21 

4.26 

4.31 

4.37 

4.42 

.48 

4.54 

.60 

4.66 

6 

85 

3.89 

3.93 

3.98 

4.02 

4.07 

4.12 

4.17 

4.22 

4.27 

4.32 

4.37 

4.43 

.48 

4.54 

4.60 

4.66 

5 

86 

3.90 

3.94 

3.98 

4.03 

4.08 

4.12 

4.17 

4.22 

4.27 

4.33 

4.38 

4.43 

.49 

4.55 

4.61 

4.67 

4 

87 

3.90 

3.94 

3.99 

4.03 

4.08 

4.13 

4.18 

4.23 

4.28 

4.33 

4.38 

4.44 

.50 

4.55 

4.61 

4.68 

3 

88 

3.90 

3.95 

3.99 

4.04 

4.08 

4.13 

4.18 

4.23 

4.28 

4.33 

4.39 

4.44 

.50 

4.56 

4.62 

4.68 

2 

89 

3.91 

3.95 

3.99 

4.04 

4.08 

4.13 

4.18 

4.23 

4.28 

4.34 

4.39 

4.44 

.50 

4.56 

4.62 

4.68 

1 

90 

3.91 

3.95 

3.99 

4.04 

4.09 

4.13 

4.18 

4.23 

4.28 

4.34 

4.39 

4.44 

.50 

4.56 

4.62 

4.68 

0 

75°  10' 

75°  20' 

75°  30' 

75°  40' 

75°  50' 

76° 

76°  10' 

76°  20' 

76°  30' 

76°  40' 

76°  50' 

77° 

77°  10' 

77°  20' 

77°  bO' 

77°  40' 

76 


U.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.    14. 


Table  of  factors  for  reduction  of  transit  observations. 

TOP  ARGUMENT=STAR'S  DECLINATION  (J). 

SIDE  ARGUMENT- STAR'S  ZENITH  DISTANCE  (C). 

[For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  opposite  page.] 


C 

77°  40" 

77°  50' 

78° 

78°  10' 

78°  20' 

78°  30' 

73°  40' 

78°  50' 

79° 

79°  10' 

79°  20' 

79°  30' 

79°  40' 

79°  50' 

80° 

C 

1 

.OS 

.08 

.08 

.08 

.09 

.09 

.09 

.09 

.09 

.09 

.09 

.10 

.10 

.10 

.10 

89 

2 

.16 

.17 

.17 

.17 

.17 

.18 

.18 

.18 

.IS 

.19 

.19 

.19 

.20 

.20 

.20 

88 

3 

.24 

.25 

.25 

.26 

.26 

.26 

.27 

.27 

.27 

.28 

.28 

.29 

.29 

.30 

.30 

87 

4 

.33 

.33 

.34 

.34 

.34 

.35 

.36 

.36 

.37 

.37 

.33 

.38 

.39 

.40 

.40 

86 

5 

.41 

.41 

.42 

.42 

.43 

.44 

.44 

.45 

.46 

.46 

.47 

.4S 

.49 

.49 

.50 

85 

6 

.49 

.50 

.51 

.51 

.52 

.52 

.53 

.54 

.55 

.56 

.56 

.57 

.58 

.59 

.60 

84 

7 

.57 

.58 

.59 

.59 

.60 

.61 

.62 

.63 

.64 

.65 

.66 

.67 

.68 

.69 

.70 

83 

8 

.65 

.66 

.67 

.68 

.69 

.70 

.71 

.72 

.73 

.74 

.75 

.76 

.78 

.79 

.80 

82 

9 

.73 

.74 

.75 

.76 

.77 

.78 

.80 

.81 

.82 

.83 

.84 

.86 

.87 

.89 

.90 

81 

10 

.81 

.82 

.84 

.85 

.86 

.87 

.88 

.90 

.91 

.92 

.94 

.95 

.97 

.98 

1.00 

80 

11 

.89 

.90 

.92 

.93 

.94 

.96 

.97 

.93 

1.00 

1.02 

1.03 

1.05 

1.06 

1.08 

1.10 

79 

12 

.97 

.99 

.00 

1.01 

1.03 

1.04 

1.00 

1.07 

1.09 

1.11 

1.12 

1.14 

1.16 

1.18 

1.20 

78 

13 

1.05 

1.07 

.08 

1.10 

1.11 

1.13 

1.14 

1.16 

1.18 

1.20 

1.22 

1.23 

1.25 

1.27 

1.30 

77 

14 

1.13 

1.15 

.16 

1.18 

1.20 

1.21 

.23 

1.25 

1.27 

1.29 

1.31 

1.33 

1.35 

1.37 

1.39 

76 

15 

1.21 

1.23 

.25 

1.26 

1.28 

1.30 

.32 

1.34 

1.36 

1.3S 

1.40 

1.42 

1.44 

1.47 

1.49 

75 

16 

1.29 

1.31 

.33 

1.34 

1.36 

1.38 

.40 

1.42 

1.44 

1.47 

1.49 

1.51 

1.54 

1.56 

1.59 

74 

17 

1.37 

1.39 

.41 

1.43 

1.45 

1.47 

.49 

1.51 

1.53 

1.56 

1.58 

1.60 

1.63 

1.66 

1.68 

73 

18 

1.45 

1.47 

.49 

1.51 

1.53 

1.55 

.57 

1.60 

1.62 

1.64 

1.67 

1.70 

1.72 

1.75 

1.78 

72 

19 

1.52 

1.54 

.57 

1.59 

1.61 

1.63 

.66 

1.68 

1.71 

1.73 

1.76 

1.79 

1.82 

1.84 

1.87 

71 

20 

1.60 

1.62 

.65 

1.67 

1.69 

1.72 

.74 

1.77 

1.79 

1.82 

1.85 

1.88 

1.91 

1.94 

1.97 

70 

21 

1.68 

1.70 

.72 

1.75 

1.77 

1.80 

1.82 

1.85 

1.88 

1.91 

1.94 

1.97 

2.00 

2.03 

2.06 

69 

22 

1.75 

1.78 

.80 

1.83 

1.85 

1.88 

1.91 

1.93 

1.% 

1.99 

2.02 

2.06 

2.09 

2.12 

2.16 

OS 

23 

1.83 

1.85 

.88 

1.90 

1.93 

1.% 

1.99 

2.02 

2.05 

2.03 

2.11 

2.14 

2.18 

2.21 

2.25 

67 

24 

1.90 

1.93 

.96 

1.98 

2.01 

2.04 

2.07 

2.10 

2.13 

2.16 

2.20 

2.23 

2.27 

2.30 

2.34 

66 

25 

1.98 

2.00 

2.03 

2.06 

2.09 

2.12 

2.15 

2.18 

2.22 

2.25 

2.28 

2.32 

2.36 

2.39 

2.43 

65 

26 

2.05 

2.08 

2.11 

2.14 

2.17 

2.20 

2.23 

2.26 

2.30 

2.33 

2.37 

2.41 

2.44 

2.48 

2.52 

04 

27 

2.12 

2.15 

2.18 

2.21 

2.24 

2.28 

2.31 

2.34 

2.38 

2.42 

2.45 

2.49 

2.53 

2.57 

2.61 

63 

28 

2.20 

2.23 

2.26 

2.29 

2.32 

2.36 

2.39 

2.42 

2.46 

2.50 

2.54 

2.58 

2.62 

2.66 

2.70 

62 

29 

2.27 

2.30 

2.33 

2.36 

2.40 

2.43 

2.47 

2.50 

2.54 

2.58 

2.62 

2.66 

2.70 

2.75 

2.79 

61 

30 

2.34 

2.37 

2.40 

2.44 

2.47 

2.51 

2.54 

2.58 

2.62 

2.66 

2.70 

2.74 

2.79 

2.83 

2.88 

60 

31 

2.41 

2.44 

2.48 

2.51 

2.55 

2.58 

2.62 

2.66 

2.70 

2.74 

2.78 

2.83 

2.87 

2.92 

2.97 

59 

32 

2.48 

2.51 

2.55 

2.58 

2.62 

2.66 

2.70 

2.74 

2.78 

2.82 

2.86 

2.91 

2.95 

3.00 

3.05 

58 

33 

2.55 

2.58 

2.62 

2.66 

2.69 

2.73 

2.77 

2.81 

2.85 

2.90 

2.94 

2.99 

3.04 

3.09 

3.14 

57 

34 

2.62 

2.65 

2.69 

2.73 

2.76 

2.80 

2.84 

2.89 

2.93 

2.98 

3.02 

3.07 

3.12 

3.17 

3.22 

56 

35 

2.68 

2.72 

2.76 

2.80 

2.84 

2.88 

2.92 

2.96 

3.01 

3.05 

3.10 

3.15 

3.20 

3.25 

3.30 

55 

36 

2.75 

2.79 

2.83 

2.87 

2.91 

2.95 

2.99 

3.04 

3.08 

3.13 

3.18 

3.23 

3.28 

3.33 

3.38 

54 

37 

2.82 

2.86 

2.90 

2.94 

2.98 

3.02 

3.06 

3.11 

3.15 

3.20 

3.25 

3.30 

3.36 

3.41 

3.47 

53 

38 

2.88 

2.92 

2.96 

3.00 

3.04 

3.09 

3.13 

3.18 

3.23 

3.28 

3.33 

3.38 

3.43 

3.49 

3.55 

52 

39 
40 

2.95 
3.01 

2.99 
3.05 

3.03 
3.09 

3.07 
3.14 

3.11 
3.18 

3.16 
3.22 

3.20 
3.27 

3.25 
3.32 

3.30 

3:37 

3.35 
3.42 

3.40 
3.47 

3.45 
3.53 

3.51 
3.58 

3.56 
3.64 

3.62 
3.70 

51 
SO 

41 

3.07 

3.11 

3.16 

3.20 

3.24 

3.29 

3.34 

3.39 

3.44 

3.49 

3.54 

3.60 

3.  66 

3.72 

3.78 

49 

42 
43 

3.13 
3.19 

3.18 
3.24 

3.22 
3.28 

3.26 
3.83 

3.31 
3.37 

3.36 
3.42 

3.41 

3.47 

3.46 
3.52 

3.51 
3.57 

3.56 
3.63 

3.61 
3.68 

3.67 
3.74 

3.73 
3.80 

3.79 
3.86 

3.85 
3.93 

48 
47 

44 

3.25 

3.30 

3.34 

3.39 

3.43 

3.48 

3.54 

3.59 

3.64 

3.70 

3.75 

3.81 

3.87 

3.94 

4.00 

46 

45 

3.31 

3.36 

3.40 

3.45 

3.50 

3.55 

3.60 

3.65 

3.71 

3.73 

3.82 

3.83 

3.94 

4.01 

4.07 

45 

77°  40' 

77°  50' 

78° 

78°  W 

78°  20" 

78°  30' 

78°  40' 

78°  50' 

79° 

79°  10' 

79°  20' 

79°  30' 

79°  W 

79'  50' 

80° 

DETERMINATION   OP   TIME. 


77 


Table  of  factors  for  reduction  of  transit  observations. 

TOP  ARGUMENT=STAR'S  DECLINATION  (}). 

SIDE  ARGUMENT- STAR'S  ZENITH  DISTANCE  «). 

[For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C  use  bottom  line  on  this  page.) 


C 

77"  40" 

77°  SO1 

78° 

78°  10' 

78°  20" 

78°  30' 

78°  40* 

78°  50' 

79° 

79°  W 

79°  20* 

79°  3W 

79°  40' 

79°  ay 

80° 

C 

40 

3.37 

3.41 

3.46 

3.51 

3.56 

3.61 

3.66 

3.71 

3.77 

3.83 

3.89 

3.95 

4.01 

4.08 

4.14 

44 

47 

3.42 

3.47 

3.52 

3.57 

3.62 

3.67 

3.72 

3.78 

3.83 

3.89 

3.95 

.01 

4.08 

4.14 

4.21 

43 

48 

3.48 

3.53 

3.57 

3.62 

3.68 

3.73 

3.78 

3.84 

3.89 

3.95 

4.02 

.08 

4.14 

4.21 

4.28 

42 

49 

3.53 

3.58 

3.63 

3.68 

3.73 

3.79 

3.84 

3.90 

3.96 

4.02 

4.08 

.14 

4.21 

4.28 

4.35 

41 

so 

3.59 

3.63 

3.68 

3.74 

3.79 

3.84 

3.90 

3.96 

4.02 

4.08 

4.14 

.20 

4.27 

4.34 

4.41 

40 

ol 

3.64 

3.69 

3.74 

3.79 

3.84 

3.90 

3.96 

4.01 

4.07 

4.14 

4.20 

.26 

4.33 

4.40 

4.48 

39 

52 

3.69 

3.74 

3.79 

3.84 

3.90 

3.95 

4.01 

4.07 

4.13 

4.19 

4.26 

.32 

4.39 

4.46 

4.54 

38 

53 

3.74 

3.79 

3.84 

3.89 

3.95 

4.01 

4.06 

4.12 

4.19 

4.25 

4.32 

.38 

4.45 

4.52 

4.60 

37 

54 

3.79 

3.84 

3.89 

3.94 

4.00 

4.00 

4.12 

4.18 

4.24 

4.30 

4.37 

4.44 

4.51 

4.58 

4.66 

36 

55 

3.83 

3.89 

3.94 

3.99 

4.05 

4.11 

4.17 

4.23 

4.29 

4.36 

4.43 

4.50 

4.57 

4.64 

4.72 

35 

56 

3.88 

3.93 

3.99 

4.04 

4.10 

.16 

.22 

4.28 

4.34 

4.41 

4.48 

4.55 

4.62 

4.70 

4.77 

34 

57 

3.93 

3.98 

4.04 

4.09 

4.15 

.21 

.27 

4.33 

4.39 

4.46 

4.53 

4.60 

4.68 

4.75 

4.83 

33 

58 

3.97 

4.02 

4.08 

4.14 

4.19 

.25 

.32 

4.38 

4.44 

4.51 

4.58 

4.65 

4.73 

4.80 

4.88 

32 

59 

4.01 

4.07 

4.12 

4.18 

4.24 

.30 

.3fi 

4.43 

4.49 

4.56 

4.63 

4.70 

4.78 

4.86 

4.94 

31 

1*1 

4.05 

4.11 

4.17 

4.22 

4.28 

.34 

.41 

4.4,7 

4.54 

4.61 

4.68 

4.75 

4.83 

4.91 

4.99 

SO 

61 

4.09 

4.15 

4.21 

4.26 

4.32 

4.39 

.45 

4.52 

4.58 

4.65 

4.72 

4.80 

•    4.88 

4.96 

5.04 

29 

62 

4.13 

4.19 

4.25 

4.31 

4.37 

4.43 

.49 

4.56 

4.63 

4.70 

4.77 

4.85 

4.92 

5.00 

5.08 

28 

63 

4.17 

4.23 

4.29 

4.35 

4.41 

4.47 

4.55 

4.60 

4.67 

4.74 

.81 

4.89 

4.97 

5.05 

5.13 

27 

64 

4.21 

4.26 

4.32 

4.38 

4.44 

4.51 

4.57 

4.64 

4.71 

4.78 

.86 

4.93 

5.01 

5.09 

5.18 

26 

65 

4.24 

4.30 

4.36 

4.42 

4.48 

4.55 

4.61 

4.68 

4.75 

4.82 

.90 

4.97 

5.05 

5.14 

5.22 

25 

66 

4.  28 

4.34 

.40 

4.46 

4.52 

4.58 

4.65 

4.72 

4.79 

4.86 

.94 

5.01 

5.09 

5.18 

5.  26 

24 

67 

4.31 

4.37 

.43 

4.49 

4.55 

4.62 

4.68 

4.75 

4.82 

4.90 

.97 

5.05 

5.13 

5.22 

5.30 

23 

68 

4.34 

4.40 

.46 

4.52 

4.58 

4.65 

4.72 

4.79 

4.86 

4.93 

5.01 

5.09 

5.17 

5.25 

5.34 

22 

69 

4.37 

4.43 

.49 

4.55 

4.62 

4.68 

4.75 

4.82 

4.89 

4.97 

5.04 

5.12 

5.20 

5.29 

5.38 

21 

70 

4.40 

4.46 

.52 

4.58 

4.65 

4.71 

4.78 

4.85 

4.93 

5.00 

5.08 

5.16 

5.24 

5.32 

5.41 

20 

71 

4.43 

4.49 

.55 

4.61 

4.68 

4.74 

4.81 

4.88 

4.96 

5.03 

5.11 

5.19 

5.27 

5.36 

5.45 

19 

72 

4.45 

4.51 

.57 

4.64 

4.70 

4.77 

4.84 

4.91 

4.98 

5.06 

5.14 

5.22 

5.30 

5.39 

5.48 

18 

73 

4.48 

4.54 

.60 

4.66 

4.73 

4.80 

4.87 

4.94 

5.01 

5.09 

5.17 

5.25 

5.33 

5.42 

5.51 

17 

74 

4.50 

4.56 

.62 

4.69 

4.75 

4.82 

4.89 

4.% 

5.04 

5.11 

5.19 

5.27 

5.36 

5.45 

5.53 

16 

75 

4.52 

4.58 

.65 

4.71 

4.78 

4.85 

4.92 

4.99 

5.06 

5.14 

5.22 

5.30 

5.38 

5.47 

5.56 

15 

76 

4.54 

4.60 

4.67 

4.73 

4.80 

4.87 

4.94 

5.01 

5.09 

5.16 

5.24 

5.32 

5.41 

5.50 

5.59 

14 

77 

4.56 

4.62 

4.68 

4.75 

4.82 

4.89 

4.96 

5.03 

5.11 

5.18 

5.26 

5.3S 

5.43 

5.52 

5.61 

13 

78 

4.58 

4.64 

4.70 

4.77 

4.84 

4.91 

4.98 

5.05 

5.13 

5.20 

5.28 

5.37 

5.46 

5.54 

5.63 

12 

79 

4.60 

4.66 

4.72 

4.79 

4.85 

4.92 

5.00 

5.07 

5.14 

5.22 

5.30 

5.39 

5.47 

5.56 

5.65 

11 

80 

4.61 

4.67 

4.74 

4.80 

4.87 

4.94 

5.01 

5.08 

5.16 

5.24 

5.32 

5.40 

5.49 

5.58 

5.67 

10 

81 

4.62 

4.69 

4.75 

4.82 

4.88 

4.95 

5.03 

5.10 

5.18 

5.26 

5.34 

5.42 

5.51 

5.60 

5.69 

9 

82 

4.64 

4.70 

4.76 

4.83 

4.90 

4.97 

5.04 

5.11 

5.19 

5.27 

5.35 

5.43 

5.52 

5.61 

5.70 

8 

83 

4.65 

4.71 

4.78 

4.84 

4.91 

4.98 

5.05 

5.13 

5.20 

5.28 

5.36 

5.45 

5.53 

5.62 

5.72 

7 

84 

4.66 

4.72 

4.79 

4.85 

4.92 

4.99 

5.06 

5.14 

5.21 

5.29 

5.37 

5.46 

5.54 

5.63 

5.73 

6 

85 

4.66 

4.73 

4.79 

4.86 

4.93 

5.00 

5.07 

5.14 

5.22 

5.30 

5.38 

5.47 

5.55 

5.64 

5.74 

5 

86 

4.67 

4.73 

4.80 

4.86 

4.93 

5.00 

5.08 

5.15 

5.23 

5.31 

5.39 

5.47 

5.56 

5.65 

5.74 

4 

87 

4.68 

4.74 

4.81 

4.87 

4.94 

5.01 

5.08 

5.16 

5.23 

5.31 

5.40 

5.48 

5.57 

5.66 

5.75 

3 

88 

4.68 

4.74 

4.81 

4.87 

4.94 

5.01 

5.09 

5.16 

5.24 

5.32 

5.40 

5.48 

5.57 

5.66 

5.75 

2 

89 

4.68 

4.74 

4.81 

4.88 

4.94 

5.01 

5.09 

5.16 

5.24 

5.32 

5.40 

5.49 

5.57 

5.66 

5.76 

1 

90 

4.68 

4.74 

4.81 

4.88 

4.94 

5.02 

5.09 

5.16 

5.24 

5.32 

5.40 

5.49 

5.58 

5.67 

5.76 

C 

77"  40' 

77°  50' 

78° 

78°  10' 

78°  20' 

78°  30' 

78°  40' 

78°  50< 

79° 

79°  10" 

79°  20' 

79°  30' 

79°  40' 

79°  50' 

80° 

PART    TI. 


THE  DETERMINATION  OF  THE  DIFFERENCE  OF  LONGITUDE  OF  TWO  STATIONS. 


INTRODUCTORY. 

The  meridian  at  Greenwich  having  been  adopted  as  the  initial  one  to  which  all  longitudes 
in  the  United  States  are  to  be  referred,  the  determination  of  the  longitude  of  a  new  station 
consists  simply  in  the  determination  of  the  difference  of  longitude  of  the  new  station  and  of 
Greenwich,  or  some  station  of  which  the  longitude  reckoned  from  Greenwich  is  known.  The 
determination  of  a  difference  of  astronomic  longitude  is  nothing  more  nor  less  than  the  deter- 
mination of  the  difference  of  the  local  times  of  the  stations.1 

There  are  three  general  methods  of  determining  longitude  now  in  use,  viz,  the  telegraphic, 
the  chronometric,  and  the  lunar. 

In  the  telegraphic  method  the  error  of  the  local  chronometer  on  local  sidereal  time  is  deter- 
mined at  each  of  the  two  stations  by  the  methods  stated  in  Part  I  of  this  publication,  and 
the  two  chronometer  times  are  then  compared  by  telegraphic  signals  sent  between  the  stations. 

In  the  chronometric  method  certain  chronometers  which  are  transported  back  and  forth 
between  the  stations  take  the  place  of  the  telegraphic  signals  and  thus  serve  merely  to  compare 
the  station  chronometers. 

In  each  of  the  lunar  methods  the  observer  at  a  station  of  which  the  longitude  is  required 
observes  the  position  of  the  moon,  or  at  least  one  coordinate  of  that  position,  and  notes  the 
local  time  at  which  his  observation  was  made.  He  may  then  consult  the  Ephemeris  and  find 
at  what  instant  of  Greenwich  time  the  moon  was  actually  in  the  position  in  which  he  observed 
it.  The  difference  between  this  time  and  the  local  time  of  his  observation  is  his  longitude 
reckoned  from  Greenwich.  One  coordinate  fixing  the  position  of  the  moon  may  be  determined 
to  serve  as  a  means  of  deriving  a  longitude  by  measuring  the  right  ascension  of  the  moon  at  a 
transit  across  the  meridian;  by  measuring  the  angular  distance  between  the  moon  and  the  sun 
or  one  of  the  four  larger  planets,  or  between  the  moon  and  one  of  the  brighter  stars  or  by 
observing  the  times  of  disappearance  and  reappearance  (immersion  and  emersion)  of  a  known 
star  behind  the  moon — the  lunar  distance  of  the  star  at  those  instants  being  the  angle  sub- 
tended by  the  moon's  radius.  In  each  case  the  Greenwich  time  at  which  the  moon  occupied 
the  position  in  which  it  was  observed  is  obtained  either  from  the  Ephemeris,  from  observations 
at  Greenwich  at  about  the  time  in  question,  or  from  similar  observations  at  some  station  of 
known  longitude. 

The  determination  of  longitude  by  wireless  telegraph  is  not  discussed  in  this  publication. 
This  method  has  been  used  to  a  certain  extent  by  some  countries  with  apparently  satisfactory 
results.  It  will  no  doubt  be  used  to  a  considerable  extent  in  the  location  of  islands  which  have 
no  cable  connections.  The  writer  believes  that  it  is  much  less  expensive  and  more  satisfactory 
at  present  to  use  the  ordinary  telegraph  lines  for  the  determination  of  longitude  for  geodetic 
purposes  within  the  United  States.  These  conditions  may  be  reversed  in  the  not  distant 
future. 

1  The  times  may  he  either  sidereal  or  mean  solar.    Usually  the  sidereal  times  are  compared  because  the  time  observations  are  nearly  always 
made  upon  stars. 

78 


DETERMINATION   OF    LONGITUDE.  79 

The  telegraphic  method1  is  the  most  accurate  known  method  of  determining  differences 
of  longitude.  It  is  always  used  in  this  Survey  for  all  longitude  determinations  in  regions 
penetrated  by  telegraph  lines,  and  is  therefore  set  forth  fully  in  this  publication. 

A  method  suitable  for  use  in  regions  not  reached  by  the  telegraph,2  is  the  chronometric 
method.  As  this  has  been  extensively  used  at  coast  stations  in  Alaska  and  will  probably 
continue  to  be  so  used  during  some  years  to  come,  it  is  also  here  treated  in  full. 

To  use  the  chronometric  method  one  must  be  able  to  travel  back  and  forth  carrying  chro- 
nometers between  the  two  stations.  The  cost  of  such  a  longitude  determination  increases  with 
increased  cost  of  travel  between  stations,  and  its  accuracy  decreases  as  the  time  required  to 
make  a  round  trip  increases.  These  facts  cause  the  chronometric  method  to  give  way  to  lunar 
methods  in  certain  comparatively  rare  situations.  The  points  at  which  the  boundary  between 
Alaska  and  British  America  (one  hundred  and  forty-first  meridian)  crosses  the  Yukon  and 
Porcupine  Rivers  were  determined  by  lunar  methods.3  Comparatively  few  such  cases  have 
occurred  in  late  years  in  this  Survey  in  which  it  was  desirable  to  resort  to  observations  upon 
the  moon  to  determine  important  longitudes.4  To  have  determined  these  longitudes  by  trans- 
portation of  chronometers  would  have  been  exceedingly  difficult  and  costly,  and  would  have 
given  results  of  a  low  order  of  accuracy,  for  there  are  more  than  a  thousand  miles  of  slow  river 
navigation  between  the  mouth  of  the  Yukon  and  either  station. 

As  the  lunar  methods  will  probably  be  used  less  and  less  with  the  lapse  of  time  and  the 
increase  of  traveling  facilities,  it  does  not  seem  desirable  to  incorporate  details  in  regard  to  them 
in  this  publication,  especially  as  such  details  would  greatly  increase  its  size.  The  computa- 
tions involved  are  long,  complex,  and  difficult.  Those  who  wish  to  study  the  lunar  methods 
are  referred  for  details  to  Doolittle's  Practical  Astronomy,  to  Chauvenet's  Astronomy,  Volume 
I,  and  to  the  American  Ephemeris  (aside  from  the  tables),  especially  to  the  pages  in  the  back 
of  each  volume  headed  "Use  of  tables." 

PROGRAM  AND  APPARATUS  OF  THE  TELEGRAPHIC  METHOD. 

During  more  than  60  years  of  its  use  by  the  Coast  and  Geodetic  Survey  the  telegraphic 
method  was  gradually  modified,  but  with  the  adoption  of  the  transit  micrometer  about  1904 
the  program  of  the  determination  of  primary  longitudes  underwent  radical  changes.  The  pro- 
gram and  apparatus  used  at  present  in  the  Survey  will  be  described  first  and  then  the  method 
formerly  used  will  be  briefly  explained. 

The  introduction  of  the  transit  micrometer  practically  eliminated  from  the  time  determina- 
tions, and  consequently  from  the  longitude  determinations,  the  large  error  which  was  known 
as  the  observer's  personal  equation.  The  program  of  longitude  observations  was  formerly 
designed  to  eliminate  the  personal  equation  from  the  results. 

GENERAL   INSTRUCTIONS    FOR    LONGITUDE    DETERMINATION   BY  THE   COAST  AND  GEODETIC 
SURVEY  WITH  TRANSIT  MICROMETERS  IN  LOW  LATITUDES  (LESS  THAN  50°). 

1.  The  observations  upon  each  star  should  be  given  unit  weight,  regardless  of  the  declina- 
tion of  the  star  and  of  whether  or  not  the  observation  of  the  transit  is  complete.  If  an  observed 
transit  is  incomplete,  only  those  observations  should  be  used  for  which  the  positions  of  the 
observing  wire  are  symmetrical  with  reference  to  the  middle  point  of  the  registration  interval 
of  the  screw;  that  is,  each  record  is  to  be  rejected  for  which  the  symmetrical  record  is  missing. 

1  The  telegraphic  method  of  determining  differences  of  longitude  was  originated  by  the  Coast  Survey  in  1846,  two  years  after  the  first  trans- 
mission of  telegraphic  messages  over  wires.    During  the  long  interval  since  that  time  the  method  has  gradually  been  brought  to  its  present  high 
state  of  perfection.    For  a  historical  note  on  this  subject  see  Appendix  No.  2,  Report  for  1897,  pp.  202-203. 

2  In  certain  cases  in  which  the  telegraph  line  is  wanting,  the  same  principles  may  be  used  with  the  substitution  of  a  flash  of  light  between  sta- 
tions in  the  place  of  the  electric  wave.    For  example,  one  might  so  determine  the  longitudes  of  the  Aleutian  Islands  of  Alaska,  the  successive  islands 
being  in  general  intervisible.    This  method  has  not,  however,  been  used  by  this  Survey.    The  cost  of  determining  longitudes  by  this  method  will 
in  general  bo  so  much  greater  than  by  the  chronometric  method  (because  of  the  many  intermediate  stations  which  will  be  required  between  distant 
stations),  as  to  more  than  offset  its  greater  accuracy. 

*  In  the  final  demarcation  of  the  boundary  between  Alaska  and  British  Columbia,  an  initial  point  on  the  one  hundred  and  forty-first  meridian 
was  determined  telegraphically,  using  transits  equipped  with  transit  micrometers.  The  telegraphic  longitude  came  within  the  range  of  three 
determinations  by  lunar  methods.  The  total  range  of  the  several  lunar  determinations  of  longitude  in  different  years  was  1.1  seconds  of  time. 

4  A  statement  of  the  results  of  these  determinations,  which  is  especially  interesting  as  showing  what  errors  may  be  expected  in  such  observa- 
tions, is  given  in  Appendix  No.  3  of  the  Report  for  1895. 


80  U.   S.   COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 

2.  The  limit  of  rejection  for  an  observation  upon  one  star  (whether  the  observed  transit  is 
complete  or  not)  is  a  residual  of  0.20  second.     No  observation  corresponding  to  a  residual  smaller 
than  this  should  be  rejected  unless  the  rejection  is  made  at  the  time  of  observation. 

3.  Each  half  set  of  time  observations  should  consist  of  observations  on  from  5  to  7  stars 
(6  preferred).     In  rare  cases  a  half  set  may  consist  of  only  four  stars.     All  of  these  are  to  be 
time  stars;  that  is,  no  azimuth  stars  are  to  be  observed.     For  the  purpose  of  this  paragraph  an 
azimuth  star  is  defined  as  one  for  which  the  azimuth  factor,  A,  is  greater  than  unity.     The  alge- 
braic sum  of  the  A  factors  in  each  half  set  should  be  kept  less  than  unity  unless  it  is  found  that 
to  secure  such  a  half  set  considerable  delays  would  be  necessary.     It  is  desirable  to  have  the 
algebraic  sum  of  the  A  factors  as  small  for  each  half  set  as  it  is  possible  to  make  it  by  the  use 
of  good  judgment  in  selecting  the  stars,  but  it  is  not  desirable  to  reduce  the  number  of  stars 
per  hour  to  be  observed  in  order  to  improve  the  balancing  of  the  A  factors,  if  said  balancing  is 
already  within  the  specified  limit. 

4.  In  selecting  lists  of  stars  to  be  observed,  one  should  endeavor  to  secure  the  maximum 
number  of  stars  per  hour  possible,  subject  to  the  conditions  of  paragraph  3  and  to  the  necessity 
of  securing  level  readings,  reversing  the  instrument,  exchanging  signals,  et  cetera.     To  observe 
the  same  stars  at  both  stations  involved  in  a  longitude  difference  is  desirable,  but  it  is  of  less 
importance  than  to  secure  rapid  observations  with  well-balanced  A  factors  in  each  half  set. 

5.  The  telescope  should  be  placed  in  the  position  "illumination  west"  for  the  first  half  set 
of  each  night  and  it  should  be  reversed  before  the  beginning  of  each  of  the  other  half  sets. 

6.  The  observations  on  each  night  should  consist,  under  normal  conditions,  of  four  such 
half  sets  as  are  defined  in  paragraph  3.     In  case  of -interference  with  the  normal  progress  of  the 
observations  by  clouds  or  other  causes,  a  determination  on  a  given  night  may  be  allowed  to 
depend  upon  a  smaller  number  of  stars  and  of  half  sets  at  each  station.     But  the  determination 
of  the  longitude  difference  on  any  night  is  to  be  rejected  if,  at  either  station,  there  has  been  no 
reversal  of  the  instrument,  or  if  less  than  twelve  stars  with  two  reversals  are  successfully 
observed  at  either  station,  or  if  the  exchange  of  signals  takes  place  at  either  station  outside  the 
interval  covered  by  the  time  observations  at  that  station. 

7.  There  is  to  be  no  exchange  of  observers  during  the  determination  of  any  difference  of 
longitude. 

8.  A  determination  of  a  difference  of  longitude  will  consist  of  either  three  or  four  such 
nights  of  observations  as  are  specified  in  paragraph  6.     If,  before  an  opportunity  occurs  to 
take  observations  upon  a  fourth  night,  it  becomes  known  that  the  result  from  each  of  the  first 
three  nights  of  observations  agrees  with  the  mean  result  within   0».070,  no  observations  on  a 
fourth  night  should  be  taken.     If  one  or  more  of  the  first  three  nights  give  results  differing  by 
0*.070  or  more  from  the  mean,  or  if    observations  are  secured  on  a  fourth  night  before  the 
results  from  the  first  three  nights  are  all  known,  then  observations  on  four  nights  are  to  con- 
stitute a  complete  determination  of  a  difference  of  longitude. 

9.  When  referring  a  longitude  station  to  a  triangulation  station  the  angle  and  distance 
measurements  should  be  made  with  a  check  and  with  such  accuracy  that  if  necessary  the 
longitude  station  may  replace  the  triangulation  station  for  future  surveys. 

10.  The  field  computations  are  to  be  kept  as  closely  up  to  date  as  practicable. 

11.  In  making  the  computations  of  time  observations  in  the  field,  the  method  shown  on 
pages  21  to  27  of  this  publication  should  be  followed. 

GENERAL  INSTRUCTIONS  FOR  LONGITUDE   DETERMINATION  BY  THE  COAST  AND  GEODETIC 
SURVEY  WITH  TRANSIT  MICROMETERS  IN  HIGH  LATITUDES  (GREATER  THAN  50°). 

The  observing  and  the  field  computations  for  the  work  in  connection  with  the  telegraphic 
determination  of  longitude  in  latitudes  greater  than  50°  should  be  done  in  accordance  with  the 
instructions  for  work  in  latitudes  less  than  50°  except  that:  (a)  The  stars  of  a  set  are  given 
different  weights  depending  upon  their  positions.  (V)  No  rejection  limit  is  fixed  for  use  by  the 
observer;  rejections  are  made,  if  necessary,  in  the  office  after  the  least  square  computations 
have  been  made,  (c)  It  will  be  impossible,  as  a  rule,  to  have  a  half  set  with  all  time  stars  and 


No.  10. 


(Chronometer      (Condenser 


rConde 


-=p-Battery 


Chronometer  Relay 


Battery  -=- 


III  Chronograph 


(Relay 


Battery 


Transit  Micrometer 


Battery  -SST 


Telegrapher's  &  Signal  Key 


Mam  Line 


During  Time  Observations 


/Chronometer       (Condenser 


Battery 


Telegrapher's  &  Signal  Key 


Battery  ~=F 


During  Exchange  of  Signals 


ARRANGEMENT  OF  ELECTRICAL  CONNECTIONS,  TELEGRAPHIC   LONGITUDE— TRANSIT-M  ICROM  ETER 

METHOD. 


No.   11. 


(•Chronometer  /-Condenser 

Vx^^x 


1 

- 


Chronometer  Relay 


Battery  -=- 


Chronograph 


I  Observing  Key 


LJ 


Battery 


+     4 


Signal  Relay 


1 
J      ^Sounder  Relay 


O  Telegn 

__ 


apher's  &  Signal  Key 


Main  Line 


During  Time  Observations 


(Chronometer        (Condenser 
yffTT^ 


Battery  -d= 


During    Exchange  of  Signals 


ARRANGEMENT  OF  ELECTRICAL  CONNECTIONS,  TELEGRAPHIC   LONGITUDE-KEY   METHOD. 


DETERMINATION   OF    LONGITUDE.  81 

hence,  the  half  sets  are  to  be  made  up  of  time  and  azimuth  stars.  (An  azimuth  star  is  one  hav- 
ing an  A  factor  greater  than  unity.)  (d)  In  making  the  computation  of  the  time  observations 
the  observer  will  use  his  discretion  as  to  the  method  to  be  used,  provided  it  is  one  of  those 
given  in  this  pubb'cation. 

USUAL  METHOD  OF  OPERATIONS. 

As  the  personal  equation  is  very  small,  if  it  exists  at  all,  it  is  not  considered  necessary  in 
determining  astronomic  longitudes  for  geodetic  or  geographic  purposes  to  have  an  exchange 
of  observers,  nor  is  it  necessary  that  a  new  station  should  be  in  a  closed  circuit. 

The  normal  determination  of  longitude  between  two  stations  using  transit  micrometers 
consists  of  three  nights'  observations  without  exchange  of  observers.  (Under  the  general 
instructions  a  fourth  night  is  sometimes  required.)  Each  night's  observations  consist  of  four 
half-sets  of  six  stars  each,  the  instrument  being  reversed  in  its  wyes  between  each  two  half-sets. 
Arbitrary  signals  are  usually  exchanged  between  the  two  stations  by  telegraph  in  the  interval 
between  the  second  and  third  half-sets.  This  places  the  arbitrary  signals,  by  which  the  chro- 
nometers at  the  two  stations  are  compared,  as  nearly  as  possible  in  the  middle  of  the  observing 
period  and  it  makes  the  longitude  determined  depend  equally  on  each  of  the  time  sets.  The 
two  observatories  must,  of  course,  be  connected  by  means  of  a  telegraph  line.  An  arrangement 
is  made  with  the  telegraph  company  for  a  direct  connection  between  the  stations,  at  the  required 
time,  on  nights  of  observation.  This  is  accomplished  by  running  wires  from  the  longitude 
stations  to  the  switchboards  of  the  local  telegraph  offices.  If  possible  the  line  should  be  without 
repeaters.  The  advisability  of  having  the  station  convenient  to  the  telegraph  office  should 
have  some  weight  in  determining  its  location.  Occasionally  the  station  may  have  to  be  con- 
nected directly  with  a  main  wire  instead  of  with  the  telegraph  office  switchboard. 

The  general  arrangement  of  the  electrical  apparatus  at  each  station  during  star  observa- 
tions and  also  during  exchange  of  signals  is  shown  in  the  diagrams  of  illustrations  Nos.  10  and 
11.  Illustration  No.  12  shows  the  actual  switchboard  and  instruments  used  in  these  operations. 
This  board  carries  an  ordinary  telegrapher's  key,  sounder  relay,  and  signal  relay,  all  of  which 
may  be  included  in  the  telegraph  circuit.  If  desired  the  signal  relay  or  the  sounder  relay  and 
key  may  be  cut  out  by  means  of  plug  switches.  The  sounder  is  worked  by  the  sounder  relay 
through  a  separate  battery.  When  the  operator  is  clearing  the  line  or  communicating  with  the 
operator  at  the  other  observatory,  the  signal  relay  is  cut  out,  and  when  signals  are  being  sent  it 
is  again  cut  in,  and  it  operates  the  pen  of  the  chronograph  through  a  separate  battery.  Thus, 
at  each  station,  when  the  signal  relay  is  on  the  main  line,  every  break  of  the  telegrapher's  key 
operates  the  two  signal  relays  and  makes  records  on  the  chronograph  sheets  at  both  stations. 
The  chronometers  being  placed  in  the  local  circuits  at  both  stations  continue  their  records  on 
the  chronograph  sheets,  the  circuits  being  break  circuits,  and  so  it  is  possible  to  read  from  the 
chronograph  sheet  at  each  station  the  chronometer  time  of  sending  and  receiving  the  arbitrary 
signals. 

The  local  circuit,  as  explained  on  page  12,  consists  of  one  principal  circuit,  the  chronograph 
circuit,  to  which  the  chronometer  circuit  and  the  transit  circuit  are  joined  through  the  points 
of  their  respective  relays.  The  observing  key,  when  used,  replaces  the  transit  circuit.  The 
chronograph  circuit,  connected  with  the  proper  binding  posts  of  the  switchboard,  includes  the 
points  of  the  signal  relay,  except  when  cut  out  by  a  plug  switch.  This  plug  is  kept  in  during 
time  observations,  and  taken  out  only  during  the  exchange  of  signals. 

A  few  minutes  before  the  time  for  exchange  of  signals  the  telegraph  operator  secures  a 
clear  line  between  stations,  ascertains  whether  the  observations  at  the  other  station  are  pro- 
ceeding successfully,  and  telegraphs  the  exact  epoch  at  which  signals  will  be  exchanged.  This 
epoch  is  arranged,  if  practicable,  not  to  interfere  with  the  star  observations  at  either  station. 
If  at  one  of  the  stations  floating  clouds  or  other  causes  are  making  it  difficult  to  get  observations 
the  observer  at  that  station  should  choose  the  epoch,  for  the  loss  of  one  or  more  stars  by  him 
might  cause  the  loss  of  a  night's  work.  When  the  epoch  arrives  the  points  of  the  signal  relay 
8136°— 13 6 


82  U.   S.   COAST  AND  GEODETIC   SURVEY  SPECIAL  PUBLICATION   NO.   14. 

are  placed  in  the  local  circuit  at  each  station  by  the  removal  of  a  plug  of  each  switchboard 
Any  break  in  the  main-line  circuit  will  now  cause  corresponding  breaks  in  the  local  circuits, 
and  a  signal  made  with  the  telegraph  key1  will  be  recorded  on  both  chronographs.  The 
observer  at  the  western  station  customarily  sends  signals  first,  by  releasing  the  telegraph 
key  for  an  instant  between  the  breaks  of  his  chronometer  at  an  average  interval  of  two 
seconds.  He  times  these  signals  so  that  they  will  not  interfere  with  his  own  chronometer 
record,  and  he  must  also  be  prepared  to  shift  them  to  another  portion  of  the  second,  if  they  are 
conflicting  with  the  record  of  the  chronometer  at  the  other  station.  Notice  of  an  interference  is 
given  by  the  other  observer  by  breaking  into  the  circuit  and  making  a  succession  of  quick 
breaks  with  the  key.  After  15  to  20  signals  have  been  sent  from  the  western  station,  covering 
a  period  of  over  half  a  minute,  double  that  number  of  signals  are  sent  by  the  eastern  observer, 
and  then  15  to  20  more  are  sent  by  the  western  observer.  This  makes  a  total  of  30  to  40  signals 
each  way,  with  the  mean  epoch  of  the  signals  from  the  two  different  directions  agreeing  closely. 
The  signals,  as  a  rule,  cover  a  total  period  of  less  than  three  minutes.  It  is  well  to  make  a 
succession  of  quick  breaks  at  the  beginning  and  end  of  each  series  of  signals.  It  is  also  desirable 
to  vary  the  position  of  each  of  several  signals  with  reference  to  the  chronometer  breaks  at  the 
beginning  of  a  series  or  to  make  several  signals  at  intervals  of  one  second  in  order  to  facilitate 
the  identification  of  corresponding  records  at  the  two  stations.  The  number  of  signals  exchanged 
is  arranged  to  cover  a  period  greater  than  one  minute  each  way,  with  a  view  of  eliminating  errors 
in  the  contact  wheel  of  the  chronometer. 

A  signal  sent  from  one  station  to  the  other  will  be  recorded  on  the  chronograph  of  the 
sending  station  slightly  before  it  is  on  the  distant  chronograph,  and  this  difference  in  time  of 
record  is  called  the  transmission  time.  It  depends,  in  fact,  both  on  the  retardation  of  the  signal 
in  the  telegraph  line  between  the  two  stations,  and  on  the  difference  in  the  time  of  action  of  the 
signal  relays  at  the  two  stations. 2  Signals  sent  from  west  to  east  will  make  the  difference  in 
longitude  too  large,  and  signals  from  east  to  west  will  make  it  too  small  by  the  amount  of  the 
transmission  time.  By  taking  the  mean  of  the  differences  as  given  by  the  signals  in  both 
directions  this  source  of  error  is  eliminated,  provided  the  transmission  time  is  the  same  in 
both  directions.3 

During  exchange  of  signals  the  chronographs  are  run  at  double  speed,  so  that  the  signals 
may  be  read  to  hundredths  of  seconds.  The  advantage  in  sending  signals  by  making  arbitrary 
breaks  of  the  circuit  is  that  they  will  come  at  varying  parts  of  the  seconds,  thus  tending  to  elimi- 
nate personal  equation  in  the  reading  of  the  fractional  parts  of  the  second.4  If  portions  of  the 
record  are  missed,  the  corresponding  signals  at  the  two  stations  may  still  be  identified  by  com- 
paring the  successive  differences  between  signals. 

RECORD  OF  AN  EXCHANGE  OF  SIGNALS. 

The  following  is  one  night's  record  of  an  actual  exchange  of  signals  between  two  stations, 
written  as  read  from  the  chronograph  sheet  on  a  special  form  used  for  the  purpose,  on  which 
is  also  made  the  computation  of  the  epochs  of  the  signals  at  the  two  stations,  the  computation 
of  the  final  difference  of  signals,  and  the  transmission  time. 

1  It  is  to  be  noted  that  these  signals  are  made  by  breaking  the  circuit,  which  is  opposite  to  the  ordinary  correspondence  use  of  the  key. 

2  The  latter  is  probably  a  small  quantity.    Some  measurements  of  the  armature  time  of  one  of  the  quick-acting  relays  used  in  these  longitude 
determinations  showed  it  to  vary  from  0.005  to  0.015  second  with  extreme  changes  in  adjustments  and  current. 

3  There  is  always  some  uncertainty  on  this  score  when  repeaters  are  used  in  the  mam  telegraph  line,  because  of  the  distinct  mechanical  arrange- 
ments for  repeating  the  signals  in  the  two  directions.    Repeaters  are  therefore  to  be  avoided  as  far  as  practicable. 

*  Chronometer  signals  were  formerly  used— that  is,  the  chronometers  were  alternately  made  to  send  their  breaks  through  the  main-line  circuit, 
recording  on  both  chronographs.  Some  of  the  objections  to  this  method  were  liability  of  damage  to  the  points  of  the  break  circuit  wheel  of  the 
chronometer  when  put  on  the  main  line,  possibility  of  the  record  of  one  chronometer  interfering  with  the  record  of  the  other,  and  personal  equation 
in  reading  a  record  that  always  occurred  at  the  same  part  of  a  second. 


D 
- 

5 

z 

o 


I 

A, 
< 
o: 
0 
u 
_i 
u 
1- 

Q 


DETERMINATION   OF   LONGITUDE. 


83 


Arbitrary  signals. 


Form  256. 
[Station,  Key  West,  Fla.    Date,  Feb.  14,  1907.    Observer,  J.  S.  Hill.    Recorder,  J.  S.  Hill.] 


From  Key  West  to  Miami 

From  Miami  to  Key  West* 

Miami  record 

Key  West 
record 

Diff. 

Miami  record 

Key  West 
record 

Difl. 

T>    m       s 

ft    TO       s 

m     s 

Ti    m       s 

It    m       > 

m      8 

6    33    35.  10 

6    27    38.  28 

5    56.82 

6    34    54.41 

6    28    57.71 

5    56.  70 

36.42 

39.63 

.79 

56.32 

59.63 

.69 

37.50 

40.70 

.80 

58.31 

29    01.  60 

.71 

38.50 

41.71 

.79 

35    00.22 

03.52 

.70 

39.45 

42.67 

.78 

02.35 

05.64 

.71 

41.47 

44.67 

.80 

04.30 

07.58 

.72 

43.43 

46.63 

.80 

06.54 

09.83 

.71 

45.50 

48.69 

.81 

08.31 

11.60 

.71 

47.50 

50.70 

.80 

10.26 

13.54 

.72 

49.58 

52.77 

.81 

12.24 

15.53 

.71 

51.60 

54.78 

.82 

14.31 

17.61 

.70 

53.57 

56.77 

.80 

16.22 

19.51 

.71 

55.65 

58.85 

.80 

18.22 

21.52 

.70 

58.18 

28    01.37 

.81 

20.28 

23.57 

.71 

34    00.51 

03.71 

.80 

24.29 

27.58 

.71 

02.52 

05.72 

.80 

26.22 

29.51 

.71 

03.67 

06.88 

.79 

28.23 

31.53 

.70 

04.77 

07.95 

.82 

30.28 

33.57 

.71 

35    42.20 

29    45.40 

.80 

31.24 

34.54 

.70 

43.50 

46.72 

.78 

32.43 

35.73 

.70 

45.08 

48.29 

.79 

34.25 

37.54 

.71 

47.50 

50.70 

.80 

49.56 

52.74 

.82 

51.50 

54.70 

.80 

53.47 

56.67 

.80 

55.64 

58.84 

.80 

57.59 

30    00.80 

.79 

59.57 

02.79 

.78 

36    01.57 

04.77 

.80 

03.58 

06.80 

.78 

05.55 

08.76 

.79 

07.55 

10.76 

.79 

09.60 

12.80 

.80 

11.53 

14.74 

.79 

12.  62 

15.85 

.77 

13.57 

16.77 

.80 

14.  61 

17.81 

.80 

15.54 

18.73 

.81 

Means: 

6    34.9 

6    29.0 

5    56.798 

6    35.3 

6    29.3 

5    56.707 

6    34.9 

6    29.0 

5    56.798 

Means 

6    35.  1 

6    29.1 

5    56.  752 

Transmission  time=  .  046 

*  Complete  set  of  signals  from  Miami  to  Key  West  not  obtained. 

In  the  foregoing  table  the  mean  epochs  are  shown  for  the  record  of  signals  on  each  chrono- 
graph sheet,  the  mean  of  all  the  differences  of  the  chronometer  records,  and  the  transmission 
time.  It  is  usually  sufficient,  in  obtaining  the  mean  epochs  of  signals,  where  they  are  symmet- 
rically arranged,  to  take  the  mean  of  the  first  five  and  the  last  five  signals. 

CHRONOMETER  CORRECTIONS  AND  RATES. 

»  On  the  following  form  are  tabulated  the  epochs  (T0)  for  which  chronometer  corrections 
were  determined  at  both  stations,  the  corrections  (AT)  determined,  and  the  rate  per  minute 
computed  from  the  two  time  sets  observed  on  each  night.  In  each  case  the  mean  of  the  epochs 
and  of  the  corrections  is  given  on  the  third  line  for  each  date.  These  means  furnish  a  correction 
for  the  chronometer  very  nearly  at  the  epoch  of  the  signals,  and  they  thus  reduce  the  work  of 
computing  the  chronometer  corrections  for  the  epochs  of  the  signals. 


84 


U.   S.   COAST   AND   GEODETIC   SUBVEY   SPECIAL   PUBLICATION    NO.   14. 

Chronometer  corrections  and  rates. 


Date 

Key  West,  Fla. 

Rate  per 
minute 

Miami,  Fla. 

Rate  per 
minute 

To 

JT 

To 

AT 

1907. 
Feb.     14 

1    m 
5    49.6 
7    11.0 
6    30.3 

s 
+14.691 
+  14.726 
+  14.708 

1 
+0.00043 

h     m 
S    41.4 
7    19.1 
6    30.2 

+45.  !77 
+45.493 
+45.335 

s 
+0.  00323 

15 

5    50.0 
7    47.9 
6    49.0 

+  14.327 
+  14.220 
+  14.274 

-0.00091 

5    46.9 
7    11.0 
6    29.0 

+50.  182 
+50.449 
+50.  316 

+0.00317 

16 

5    50.1 

7    09.0 
6    29.5 

+  13.479 
+  13.460 
+  13.470 

-0.00024 

5    54.7 
7    22.4 
6    38.6 

+55.337 
+55.551 
+55.444 

+0.00244 

COMPUTATION  OF  DIFFERENCE  OF  LONGITUDE. 

The  next  step  is  the  computation  of  the  difference  of  longitude  from  the  mean  of  the  signals 
sent  in  each  direction.  Each  night's  observations  represents  a  complete  determination  of  this 
difference,  and  a  separate  and  complete  computation  is  accordingly  made  for  each  night.  The 
epoch  of  signals  and  difference  of  chronometers  are  taken  from  the  record  of  signals  for  each 
night,  and  the  chronometer  corrections  at  these  epochs  are  computed  for  each  station  and  each 
night,  using  the  rates  per  minute  given  in  the  preceding  form.  To  the  difference  in  chronometers 
is  then  applied  the  difference  in  chronometer  corrections  (eastern  minus  western  chronometer), 
which  gives  the  difference  of  longitude  in  time  as  determined  by  the  night's  observations. 
From  this  determination  the  transmission  time  has  already  been  eliminated  by  taking  the  means 
of  eastern  and  western  signals. 

The  chronometer  correction  A  T  at  the  time  of  exchange  T  and  its  probable  error  r  are 
expressed  by 

-  71),  and  r 


where  AT^  and  ±r\  are  the  chronometer  correction  and  its  probable  error  derived  from  the 
first  set  of  time  observations  at  epoch  Tl}  and  AT2  and  ±r2  are  the  same  quantities,  respec- 
tively, for  the  second  set  at  epoch  T2. 

Computation  of  difference  of  longitude. 

BETWEEN  MIAMI  AND  KEY  WEST,  FLA. 


T0 

AT 

Date 

Diff:  JT 

Difl.  Of 
signals 

Ji 

V 

Trans- 
mission 
time 

Miami 

Key  West 

Miami 

Key  West 

1907. 

h     m 

h     m 

3 

s 

s 

m      s 

m      s 

s 

s 

Feb.  14 

6    35.1 

6    29.1 

+45.351 

+14.709 

+30.642 

5    56.752 

6    27.394 

-0.031 

0.046 

15 

6    31.7 

6    25.8 

+50.325 

+  14.295 

+36.030 

5    51.285 

6    27.  315 

+  .048 

.051 

16 

6    33.6 

6    27.8 

+55.432 

+  13.470 

+41.962 

5    45.418 

6    27.380 

-  .017 

.047 

Mean.. 

6    27.363 

Reduction  to  longitude  pier  of  1896=  .97  meter 
Reduction  to  mean  position  of  pole ' 


=  +0.002 
-     0.000 

m     t 
Miami  longitude  station  east  of  Key  West  longitude  station-  6    27.365 

=  1°36'50".525 


In  the  example  shown  above  the  second  column  gives  the  mean  epoch  of  the  exchange 
of  signals  as  read  from  the  chronograph  sheet  at  the  eastern  station,  Miami,  and  the  fourth 
column  gives  the  correction  to  the  chronometer  at  Miami  for  the  mean  epoch  of  the  signals, 
this  correction  being  computed  from  the  corrections  to  the  chronometer  and  the  rate  deduced 
from  the  time  observations.  The  third  and  the  fifth  columns  give  similar  data  for  the  western 


i  See  Astronomische  Nachrichten  No.  4253. 


DETERMINATION   OF   LONGITUDE. 


85 


station,  Key  West.  The  difference  between  the  chronometer  corrections  (AT)  given  in  the 
fourth  and  fifth  columns  is  shown  in  the  sixth  column  and  equals  the  correction  at  the  eastern 
station  minus  the  correction  at  the  western  station.  In  the  next  column  is  given  the  difference 
of  signals  (eastern  minus  western).  The  difference  of  longitude,  AX,  is  then  the  combination 
of  the  difference  between  the  A  T's  at  the  two  stations  and  the  difference  of  signals.  The  trans- 
mission time  is  taken  from  the  form  on  which  the  record  of  signals  and  their  reduction  is  shown, 
and  is  placed  in  the  last  column,  while  in  the  column  immediately  preceding  is  placed  the  differ- 
ence between  each  night's  determination  and  the  mean  of  the  determinations  of  all  the  nights. 
The  values  from  the  various  nights  are  each  given  unit  weight,  and  their  mean  is  then 
considered  to  be  the  observed  difference  of  longitude  between  the  transit  instruments  at  the 
two  stations.  In  the  example  given  this  difference  has  a  correction  applied  to  it  to  reduce  it 
to  what  it  would  have  been  had  the  transit  at  the  base  station,  Key  West,  been  placed  exactly 
over  the  position  occupied  by  the  transit  in  1896  (adjusted  in  the  longitude  net  of  the  United 
States)1  instead  of  at  a  position  0.97  meters  east  of  it.  The  particular  example  given  is  one  of 
a  series  of  differences  of  longitude  determined  in  1907,  commencing  at  Key  West  and  closing 
on  Atlanta.  There  is  also  at  the  latter  place  an  adjusted  longitude  station  of  the  longitude 
net  of  the  United  States.  The  longitudes  of  these  two  stations,  at  Key  West  and  Atlanta, 
being  held  fixed,  a  closing  discrepancy  was  developed  which  was  distributed  equally  among  the 
various  differences,  each  difference  being  given  unit  weight.  The  following  table  shows  the 
differences  of  longitude  determined  between  Key  West  and  Atlanta  and  the  distribution  of 
the  closing  error: 

Computation  of  closing  error  between  Key  West  and  Atlanta. 


Observed 
difference 


Miami  west  of  Key  West 
Jupiter  west  of  Miami 
Sebastian  west  of  Jupiter 
Daytona  west  of  Sebastian 
Fernandina  west  of  Daytona 
Atlanta  west  of  Fernandina 

Atlanta  west  of  Key  West 
Atlanta  west  of  Key  West 

(From  adjusted  longitude  net  of  United  States) 


m 

-  6 

-  0 
+  1 
+  2 
+  1 
+11 


27.  365 
27.404 
33.  654 
11.  332 
46.  878 
42.  609 


+  10 
+10 


19.704 
19.  759 


Correc- 
tion to 
close 
circuit 


+  .009 
+.009 
+.009 
+.009 
+.009 
+.010 

+.055 


Adjusted 
difference 


m 

-  6 

-  0 

+  1 
+  2 
+  1 
+11 


27.  356 
27.  395 
33.  663 
11. 341 

46.  887 
42.  619 


+10    19. 759 


Closing  error=     + 


.055 


CORRECTION  FOR  VARIATION  OF  THE  POLE. 

v  A  correction  is  necessary  to  reduce  the  observed  astronomic  longitude  to  the  mean  posi- 
tion of  the  pole.  About  the  middle  of  each  year  the  Latitude  Service  of  the  International 
Geodetic  Association  publishes  in  the  Astronomische  Nachrichten  provisional  values  of  the 
coordinates  of  the  instantaneous  pole  for  the  preceding  calendar  year,  together  with  tables  to 
reduce  observed  latitudes,  longitudes,  and  azimuths  to  the  mean  position  of  the  pole.  The 
proper  correction  to  the  longitude  may  be  computed  by  means  of  these  tables,  knowing  the 
time  of  observation  and  the  latitude  and  longitude  of  the  observing  station. 

DISCUSSION  OF  ERRORS  WHEN  TRANSIT  MICROMETER  IS  USED. 

Let  it  be  supposed  that  the  regular  program  for  observations  with  a  transit  micrometer, 
three  nights'  observations  without  exchange  of  observers,  has  been  carried  out.  The  computed 
result,  the  difference  of  astronomic  longitude  of  the  two  places,  is  subject  to  the  following 
errors : 


1  See  Appendix  2  of  the  Report  for  1897. 


86  U.  S.   COAST  AND  GEODETIC   SURVEY  SPECIAL  PUBLICATION   NO.  14. 

First.  An  accidental  error  arising  from  the  accidental  errors  of  observations  of  about  T2 
stars  at  each  station.  If  the  accidental  error  of  observation  of  a  single  star  be  estimated  at 
±8.07,  which  may  be  considered  sufficiently  large  to  cover  both  the  observer's  errors  and  those 
instrumental  errors  which  belong  to  the  accidental  class,  then  the  probable  error  of  the  final 
result  arising  from  this  cause  would  be  ±s.07-n  V36=  ±s.012. 

Second.  An  accidental  error  arising  from  the  accidental  errors  in  the  adopted  right  ascen- 
sions of  such  stars  as  are  observed  at  one  station  on  a  given  night  but  not  at  the  other.  It 
is  in  such  cases  only  that  errors  in  right  ascension  have  any  effect  on  the  computed  result.  If 
entirely  different  stars  were  observed  at  the  two  stations,  24  at  each  station,  and  if  ±s.03  be 
accepted  as  the  probable  error  of  a  right  ascension,  then  the  probable  error  of  the  result  for  one 
night  arising  from  this  source  would  be  ±8.03^  V12  =  ±s-009.  In  ordinary  cases,  in  which  the 
number  of  stars  not  common  to  both  stations  is  less  than  10  per  cent,  this  accidental  error  is 
reduced  to  less  than  ±8.001. 

Third.  Errors  due  to  the  assumption  that  the  rate  of  the  chronometer  is  constant  during 
and  between  the  two  time  sets  of  a  night.  As  the  interval  between  the  mean  epochs  of  the 
sets  is  ordinarily  only  about  one  hour,  these  errors  are  probably  exceedingly  small.  In  order 
to  make  these  errors  inappreciable,  longitude  observers  should  use  chronometers  known  to 
show  but  small  variations  in  rate,  and  should  protect  them  as  thoroughly  as  is  feasible  while  in 
use  against  jars  and  sudden  changes  of  temperature.  The  errors  from  this  source  will  be  of 
about  the  same  value  whether  the  exchange  of  signals  is  made  at  about  the  mean  epoch  of  the 
two  sets  of  time  observations,  or  is  made  at  any  other  epoch  within  the  interval  covered  by  the 
two  sets. 

Fourth.  The  question  of  the  personal  equation  with  the  transit  micrometer  is  discussed 
fully  on  pages  90  and  91. 

Fifth.  Errors  arising  from  lateral  refraction.  The  probable  minuteness  of  these  errors 
in  time  observations  has  already  been  commented  upon  (see  p.  48).  It  is  not  impossible, 
however,  that  small  constant  errors  may  arise  from  this  source  at  stations  established  in  closely 
built-up  portions  of  great  cities,  particularly  of  manufacturing  centers. 

Sixth.  Errors  arising  from  variation  of  transmission  time.  By  transmission  time  is 
meant  the  interval  that  elapses  from  the  instant  at  which  the  signal  relay  breaks  the  local 
circuit  at  the  sending  station  to  that  at  which  the  signal  relay  breaks  the  local  circuit  at  the 
receiving  station.  This  interval  is  made  up  of  armature  time,  induction  time,  and  the  true 
transmission  time  of  the  electric  wave  passing  along  the  wire.  It  is  only  the  variation  in 
transmission  time  occurring  during  the  exchange  of  signals  on  each  night  that  introduces  error 
into  the  computed  result.  As  this  interval  is  not  much  over  a  minute  the  error  is  probably 
insensible  if  there  is  a  continuous  wire  connection  between  stations.  If  the  line  between 
stations  passes  through  a  "repeater"  the  transmission  time  in  one  direction  through  the 
repeater  will  be  different  from  that  in  the  other  direction  unless  the  two  magnets  of  the  repeater 
are  adjusted  exactly  alike,  and  half  this  difference  will  enter  into  the  computed  result  as  an  error. 
The  repeaters  used  in  ordinary  telegraph  service  are  not  specially  designed  for  quick  action, 
as  are  the  signal  relays  on  the  Coast  and  Geodetic  Survey  switch  board,  nor  is  their  adjustment 
in  the  control  of  the  longitude  observers.  Hence  the  desirability  of  a  continuous  wire 
connection. 

Any  change  in  transmission  time  within  the  local  circuit  during  the  exchange  of  signals 
will  produce  an  error  in  the  computed  longitude,  but  such  changes  are  probably  insensible. 
A  change  at  any  other  time  in  the  local  circuit  will  appear  in  the  observations  as  a  change  in 
the  chronometer  correction  and  will  probably  have  no  appreciable  effect  on  the  final  result 
for  the  night. 

Seventh.  The  difference  of  the  transmission  time  through  the  two  signal  relays  and  also 
the  difference  in  the  transmission  time  through  the  two  transit  micrometer  relays  enter  as 
errors  in  the  final  result.  These  errors  are  made  very  small  in  the  present  longitude  work  of 
the  Survey  by  using  relays  which  are  as  nearly  alike  as  can  be  made,  and  which  are  specially 
designed  to  act  very  quickly. 


DETERMINATION   OF  LONGITUDE. 


87 


If  the  difference  of  longitude  which  is  being  measured  is  large,  it  becomes  necessary  to 
abandon  the  practice  of  observing  the  same  stars  at  both  stations  in  order  to  make  the  exchange 
of  arbitrary  signals  come  within  the  period  of  the  night's  observations  at  each  station.  How- 
ever, the  errors  of  right  ascension  thus  introduced  will  not  be  large. 

The  combination  of  the  numerical  values  of  the  above  errors  will  not  fully  account  for  the 
error  of  the  result  as  computed  from  the  separate  determinations,  that  is  from  the  residuals, 
but  it  may  be  that  some  of  the  above  errors  for  which  no  numerical  values  are  estimated  are 
much  larger  than  supposed.  The  discussion  of  errors  of  time  observations  on  pages  48-51 
of  this  publication  applies  to  a  certain  degree  to  longitude  work. 

See  also  Discussion  of  Errors,  when  the  key  method  is  used,  on  page  93. 

PROGRAM  WHERE   NO  TRANSIT  MICROMETER   IS  USED. 

Before  the  adoption  of  the  transit  micrometer  for  longitude  work,  when  the  chronograph 
and  key  method  was  in  use,  it  was  necessary  in  all  determinations  of  differences  of  longitude  to 
arrange  the  program  of  observations  so  as  to  eliminate  the  personal  equation  of  the  observers 
making  the  time  observations.  The  personal  equation  was  eliminated  either  directly  by 
exchange  of  observers,  or  indirectly  by  supplementary  observations,  themselves  independent 
of  the  longitude  observations,  but  which  gave  a  value  for  the  personal  equation  to  be  introduced 
into  the  computations.  Further  on,  page  90,  the  question  of  personal  equation  and  its  deter- 
mination will  be  more  fully  discussed. 

In  the  determination  of  primary  differences  of  longitude  the  personal  equation  was  elimi- 
nated by  the  observers  exchanging  stations  when  one-half  of  the  observations  had  been  made. 
One-half  the  sum  of  the  mean  determinations  before  and  after  exchange  of  observers  gave  a 
resulting  difference  of  longitude  which  was  independent  of  the  personal  equations  of  the 
observers  provided  these  personal  equations  remained  constant.  Except  for  this,  the  program 
of  observations  was  the  same  as  for  observations  with  a  transit  micrometer  (see  p.  81). 

The  arrangement  of  the  telegraphic  apparatus  was  the  same  as  described  on  page  81.  The 
observing  key  took  the  place  of  the  relay  points  of  the  transit  micrometer.  Illustration  No.  11 
shows  the  arrangement  of  the  local  and  main  circuits  while  time  observations  were  being  made, 
and  also  while  signals  were  being  exchanged.  The  switchboard  is  the  same  as  used  in  transit 
micrometer  observations,  and  is  shown  in  illustration  No.  12.  The  following  records  and 
computations  show  the  various  steps  in  observing  and  computing  an  actual  difference  of  longitude. 

Record  of  exchange  of  signals,  and  computation  of  difference  of  chronometers. 

[Station,  Atlanta,  Ga.    Date,  Mar.  7, 1896.    Observer,  G.  R.  P.    Recorder,  G.  R.  P.] 
ARBITRARY  SIGNALS. 


From  Atlanta  to  Key  West 

From  Key  West  to  Atlanta 

Key  West  record 

Atlanta  record 

Difference  of 
chronometers 

Key  West 
record 

Atlanta  record 

Difference  of 
chronometers 

h  m     s 
7  35  59.  97 
36  01.  90 
04.03 

05.96 

* 

h  m     s 
7  25  42.  39 
44.30 

46.47 

48.39 

* 

TO        S 
10  17.58 
.60 
.56 

.57 

# 

h  m     s 
7  37  08.  76 
10.82 
12.78 

15.28 
* 

h  m     s 
1  26  51.  51 
53.60 
55.52 

58.04 

# 

TO        S 
10  17.  25 
.22 
.26 

.24 

* 

# 

* 

* 

# 

* 

* 

56.90 
58.91 

26  39.  30 
41.34 

.60 
.57 

38  38.  48 
40.60 

28  21.  21 
23.33 

.27 
.27 

h    m 
Means  7  36.  5 

h    m 
7  26.2 

TO         S 

10  17.  570 

h    m 
7  37.9 

h    m 
7  27.6 

TO           S 

10  17.  249 

U.   S.   COAST   AND  GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 
SUMMARY  OF  RESULTS  OF  TIME  DETERMINATIONS  AT  ATLANTA. 


Azimuth 

Date 

Epoch  (by 
face  of  chro- 
nometer) 

Chronometer 
correction 
J7V 

Rate  per 
minute 

Collimation 

West 

East 

1896. 

h     m 

S 

s 

S 

S 

S 

Mar.      7 

6  56.4 

-13.546 

+.  00261 

+.03 

-.154 

+  .035 

7 

8  12.6 

-13.347 

-.01 

-.036 

-.070 

8 

6  56.3 

-  7.  742 

+.  00310 

-.01 

+.115 

+.089 

8 

8  12.5 

-  7.506 

-.06 

+.190 

+.313 

# 

* 

* 

# 

# 

* 

# 

* 

* 

* 

-X- 

* 

* 

ft 

27 

8  12.6 

-12.  660 

+.00043 

-.18 

+.183 

4-.  155 

27 

9  22.4 

-12.630 

-.22 

+.378 

+.167 

SUMMARY  OF  RESULTS  OF  TIME  DETERMINATIONS  AT  KEY  WEST. 


Azimuth 

Date 

Epoch  (by 
face  of  chro- 
nometer) 

Chronometer 
correction 

tr, 

Rate  per 
minute 

Collimation 

West 

East 

1896. 

h     m 

s 

s 

S 

S 

S 

Mar.      7 

6  56.4 

-11.  157 

-.00232 

-.05 

-1.108 

-1.  236 

7 

8  12.6 

-11.334 

-.03 

-1.220 

-1.108 

8 

6  56.4 

-13.994 

-.00227 

-.06 

-1.  649 

-1.447 

8 

8  12.6 

-14.  167 

-.03 

-1.644 

-1.580 

# 

# 

* 

# 

tt 

# 

# 

* 

* 

* 

# 

* 

# 

* 

27 

8  12.4 

-  4.992 

-.00223 

-.06 

-0.  181 

-0.  256 

27 

9  21.8 

-  5.147 

-.00 

-0.  121 

-0.144 

FROM  WESTERN  OR  ATLANTA  SIGNALS.* 


Date 

Epoch  of  signalsf 

Difference 
of  chronome- 
ters 

Chronometer  corrections 

JV 

(from  western 
signals) 

Key  West 
*J 

Atlanta 
TW 

Key  West 

"M 

Atlanta 

*TW 

Difference 
J  TE-J  Tw 

1896. 
Mar.      7 

8 

* 

h    m 
7  36.5 

7  36.6 

* 

h    m 

7  26.2 

7.  26.  2 
* 

m        s 
10  17.570 

10  26.  199 
it 

s 
-11.  250 

-14.085 

* 

s 
-13.468 

-  7.649 

* 

S 
+  2.  218 
-6.436 

# 

m         s 
10  19.  788 

19.  763 

* 

# 

* 

# 

•* 

* 

# 

# 

* 

27 

8  51.3 

8  41.1 

10  12.507 

-  5.079 

-12.648 

+7.  569 

20.  076 

*  Unconnected  for  transmission  time  and  personal  equation, 
f  By  face  of  chronometer. 


FROM  EASTERN  OR  KEY  WEST  SIGNALS.* 


Date 

Epoch  of  signals  f 

Difference 
of  chronome- 
ters 

Chronometer  corrections 

(from  eastern 
signals) 

Key  West 

Atlanta 

TW 

Key  West 

Atlanta 
J7V 

Difference 

1896. 
Mar.      7 

8 

* 

h    m 
7  37.9 

7  37.6 

* 

h    m 
7  27.6 
7  27.2 

* 

m        s 
10  17.  249 

10  25.  881 

* 

S 
-11.  253 

-14.087 

* 

S 

-13.464 

-  7.646 

* 

S 

+  2.  211 

-6.441 

* 

m        s 
10  19.460 
.440 

# 

* 

* 

* 

# 

* 

* 

* 

27 

8  53.3 

8  43.1 

10  12.  136 

-  5.083 

-12.  647 

+7.  564 

.700 

*  Unoorrected  for  transmission  time  and  personal  equation. 
t  By  face  of  chronometer. 


DETERMINATION   OF    LONGITUDE. 


89 


COMBINATION   OF  LONGITUDE   RESULTS. 

At  one  time  it  was  the  custom  in  the  Coast  and  Geodetic  Survey  to  combine  the  resulting 
differences  of  longitude  for  the  various  nights'  observations  by  deducing  weights  and  assigning 
them  to  the  various  values.  This  custom  is  not  now  practiced  where  transit  micrometers  are 
used,  nor  is  it  followed  where  an  accepted  program  is  carried  out  even  if  no  micrometers  are 
used.  If  a  regular  program  is  carried  out  the  various  nights'  determinations  are  given  equal 
weight,  and  direct  means  are  taken  for  the  final  value  of  the  difference  of  longitude.  How- 
ever, the  following  discussion  of  the  combination  of  longitude  results  where  the  different  nights' 
observations  are  assigned  different  weights  is  given  here  as  occasion  might  arise  where  the 
information  would  be  of  value. 

The  following  table  gives  the  collection  of  the  results  for  the  different  nights  and  their 
combination  to  develop  and  eliminate  the  transmission  time  and  personal  equation.  The 
mean  of  the  differences  of  longitude  as  derived  from  the  western  and  eastern  signals  will  be 
free  from  the  transmission  time,  and  their  difference  is  double  the  transmission  time.  The  rela- 
tive weights  for  the  resulting  differences  of  longitude  for  different  nights  are  derived  from  the 

expression  p  =  f_2  ,  where  pl  and  p3  are  the  weights  of  the  determinations  of  the  chronom- 
eter corrections  at  the  epoch  of  exchange  of  signals  at  the  two  stations,  respectively, 

or  p!  =  —  and  pt  =  — -2  in  which  ^  and  r2  are  the  probable  errors  of  the  chronometer  corrections. 

r\~  r2 

To  obtain  the  personal  equation  the  weighted  means  are  taken  for  each  position  of  the  observers, 
and  half  their  difference  is  the  personal  equation  to  be  applied  with  opposite  signs  to  the  two 
groups.  This  gives  the  corrected  result  for  difference  of  longitude  for  each  night,  and  the 
weighted  mean  of  all  the  nights  is  the  final  difference  of  longitude.  The  probable  error  of  the 

latter  is  0.674-»/v— ^j^y  v~  where  n  is  the  number  of  nights  of  observation  and  2  is  the  number  of 

unknowns  (longitude  and  personal  equation).  In  the  table  the  means  in  the  seventh  and  ninth 
columns  are  weighted  means. 

The  personal  equation  is  one-half  the  difference  in  the  weighted  results  for  the  two  posi- 
tions of  the  observers,  or 


the  sign  indicating  that  S  observes  later  than  P.     The  probable  error1  of  the  personal  equa- 
tion may  be  taken  as  identical  with  that  of  the  resulting  difference  of  longitude. 

The  transmission  time,  as  stated,  is  one-half  the  difference  between  the  results  from  western 

338 
and  eastern  signals,  or  in  this  example,  =  ~o~  =  s- 169,  an  unusually  large  value,  due  to  the 

marine  cable,  between  Key  West  and  the  mainland. 

Table  of  resulting  difference  of  longitude  between  Atlanta,  Ga.,  and  Key  West,  Flo. 


Date 

Observer 
at— 

From 
western  or 
Atlanta 
signals 
«J 

From 
eastern  or 
Key  West 
signals 

«s 

Double 
trans- 
mission 
time 
Mw 
-"" 

Mean  of 
W.andE. 
signals 

Personal 
equation 

Difference 
of 
longitude 

Combi- 
nation 
weight 
P 

Resid- 
uals 

V 

A 

KW 

1896. 
Mar.     7 
8 
9 
13 
14 

Mar.  20 
21 
25 
26 
27 

P. 
P. 
P. 
P. 

P. 

S. 
S. 
S. 

s. 
s. 

S. 
8. 
8. 
S. 
S. 

P. 
P. 
P. 
P. 

P. 

TO          S 

10  19.  788 
.763 
.754 
.802 
.842 

10  20.018 
.075 
.102 
.074 
.076 

m      s 
10  19.  460 
.440 
.445 
.495 
.522 

Mean 

10  19.686 
.705 
.737 
.721 
.700 

Mean 

a 
0  328 
.323 
.309 
.307 
.320 

m       « 
10  19.  624 
.602 
.600 

.648 
.682 

1 
+0.120 

-0.120 

m      s 
10  19.744 
.722 
.720 
.768 
.802 

.732 
.770 
.799 
.777 
.768 

S 
11 
4 
13 
21 

9 
5 
4 
8 
6 

s 
-.021 
-.043 
-.045 
+.003 
+.037 

-.033 
+.005 
+  .034 
+.012 
+  .003 

.317 

10  19.  645 

.332 
.370 
.365 
.353 
.376 

10  19.  852 
.890 
.919 
.897 
.888 

0.359 

10  19.884 

10  19.765 

±0.007 

1  Practically  the  same  result  is  obtained  by  deriving  separate  values  for  the  personal  equation  by  comparing  each  result  in  the  first  position  of 
the  observers  with  the  corresponding  result  in  the  second  position  and  computing  the  probable  error  from  the  variations  in  these  separate  values. 


90  TJ.  S.   COAST  AND  GEODETIC   SURVEY  SPECIAL  PUBLICATION   NO.   14. 

The  above  formulae  and  forms  are  used  in.  the  office  computation.  The  field  computation 
differs  from  that  made  in  the  office  in  that  the  time  computation  is  made  by  an  approximate 
field  method  shown  on  page  26  or  page  34  instead  of  the  least  square  method  given  on  page  41, 
and  that  in  the  field  no  probable  errors  or  weights  are  computed  and  indiscriminate  means  are 
taken  instead  of  weighted  means.  In  the  past  some  of  the  forms  used  in  the  field  have  been 
slightly  different  from  those  shown  above.  The  office  computation  will  be  facilitated  by  making 
the  field  computation  as  here  indicated. 

PERSONAL  EQUATION. 

The  absolute  personal  equation  in  time  observations  with  a  transit  is  the  interval  of  time 
from  the  actual  instant  of  transit  of  a  star  image  across  a  line  of  the  diaphragm  to  the  instant  to 
which  the  transit  is  assigned  by  the  observer.  When  the  time  is  observed  using  a  chronograph 
and  an  observing  key  the  absolute  personal  equation  is  simply  the  time  required  for  the  nerves 
and  the  portions  of  the  brain  concerned  in  an  observation  to  perform  their  functions.  In  the 
case  of  observations  by  the  eye  and  ear  method  the  mental  process  becomes  more  involved, 
and  the  personal  equation  depends  on  a  much  more  complicated  set  of  physical  and  psychological 
conditions  than  when  the  observations  are  made  with  a  key  and  chronograph. 

Although  the  personal  equation  has  been  studied  by  many  persons  and  for  many  years, 
little  more  can  be  confidently  said  in  regard  to  the  laws  which  govern  its  magnitude  than  that 
it  is  a  function  of  the  observer's  personality,  that  probably  whatever  affects  the  observer's 
physical  or  mental  condition  affects  its  value,  that  it  tends  to  become  constant  with  experience, 
that  it  probably  differs  for  slow  moving  and  fast  moving  stars,  and  that  it  is  different  for  very 
famt  stars  which  the  observer  sees  with  difficulty  from  what  it  is  for  stars  easily  seen. 

A  systematic  error  may  be  present  which  is  due  to  the  tendency  of  the  observer  to  place 
the  wire  always  to  the  right  or  to  the  left  of  the  center  of  the  star's  image.  This  tendency  is 
due  to  the  delects  in  the  observer's  eye  and  the  error  resulting  is  called  the  bisection  error.  At 
some  astronomic  observatories  a  reversing  prism  is  used  which  reverses  the  image  of  the  star 
midway  in  the  observations.  Thus,  during  one  half  of  the  observations  the  wire  would  be 
placed  too  far  east  and  during  the  other  half  too  far  west  of  the  center  of  the  star's  image  (or 
vice  versa)  and  the  mean  of  all  the  observations  would  be  free  from  a  bisection  error.  No 
numerical  values  are  available  for  the  effect  of  the  bisection  error  but  it  is  known  to  be  so  small 
that  it  may  be  neglected  in  all  time  and  longitude  work  for  the  usual  geodetic  and  geographic 
purposes.  (See  remarks  under  the  Description  of  the  Zenith  Telescope  on  p.  105.) 

There  are  various  mechanical  devices  for  the  determination  of  the  absolute  personal  equation 
of  an  observer,  but  as  these  are  seldom  used  they  will  not  be  discussed  here. 

The  relative  personal  equation  of  two  observers  is  the  difference  of  their  absolute  equations. 

When  observing  time  with  a  transit  micrometer  the  personal  equation,  if  any,  may  be  neg- 
lected. The  observing  does  not  consist  of  a  series  of  independent  consecutive  operations,  but 
rather  of  a  continuous  performance,  the  star's  image  being  bisected  by  the  micrometer  wire 
before  the  record  is  begun  and  kept  bisected  till  after  the  record  is  ended. 

In  Appendix  8  of  the  Report  for  1904,  entitled  "A  Test  of  the  Transit  Micrometer,"  it  was 
shown  that  if  there  is  an  actual  personal  equation  in  observing  star  transits  with  a  transit 
micrometer  it  is  so  small  as  to  be  masked  by  the  other  errors  of  observation.  Viewed  in  the 
light  of  several  years  of  actual  longitude  observations  with  the  transit  micrometer  this  conclusion 
is  fully  justified.  These  longitude  observations  involved  four  simple  or  compound  loop  closures, 
and  one  determination  with  exchange  of  observers.  In  observing  differences  of  longitude  to 
close  a  loop  the  same  observer  always  kept  in  front  as  the  work  progressed  around  the  loop,  thus 
introducing  into  the  loop  closure  an  accumulation  of  any  relative  personal  equation  that  might 
exist. 

In  1906  four  differences  determined  with  the  transit  micrometer  between  Seattle,  Wash., 
and  the  point  where  the  one  hundred  and  forty-first  meridian  boundary  of  Alaska  intersects 
the  Yukon  River,  were  combined  with  certain  Canadian  results  to  form  a  loop,  and  the  loop 
closure  was  reduced  to  zero  by  applying  a  correction  of  only  0.008  second  to  each  observed 
difference  of  longitude. 


DETERMINATION   OF    LONGITUDE.  91 

In  Texas  in  1906  the  three  differences  of  longitude  between  the  three  points,  Austin,  Alice, 
and  Isabel,  were  determined,  using  transit  micrometers  and  a  program  as  indicated  above. 
This  would  introduce  into  the  closure  three  times  any  relative  personal  equation  of  the  observers. 
The  loop  closure  was  0.038  second,  making  necessary  corrections  on  the  three  differences  of 
0.8013,  0.8013,  and  0.S012. 

In  1907  a  series  of  longitude  differences  was  determined,  using  transit  micrometers,  between 
Key  West  and  Atlanta,  for  both  of  which  stations  adjusted  values  are  given  in  the  longitude 
net  of  the  United  States,1  and  these  adjusted  values  were  held  fixed.  Six  longitude  differences 
between  these  two  stations  were  determined  in  such  a  way  as  to  accumulate  any  relative  personal 
equation  between  the  two  observers.  The  results  are  shown  on  page  85.  The  correction 
required  to  be  applied  to  each  observed  difference  to  close  the  loop  was  0.8009.  A  second  loop, 
closing  on  one  of  the  links  of  the  first  loop  or  forming  with  all  but  the  last  difference  of  the  first 
loop  a  new  loop  of  eight  links  between  the  fixed  stations,  Key  West  and  Atlanta,  obtained 
corrections  of  only  0.8008  per  link  to  close.  The  corrections  in  both  loops  were  of  the  same  sign. 

Later  in  1 907  a  series  of  longitude  differences  was  determined  in  Minnesota,  Dakota,  Nebraska, 
and  Iowa,  using  the  transit  micrometer.  The  points  held  fixed  were  the  stations  of  the  longi- 
tude net  at  Bismarck  and  Omaha.  There  were  four  condition  equations  and  ten  unknowns 
involved  in  the  adjustment  of  this  secondary  net.  -  The  largest  correction  to  an  observed  differ- 
ence of  longitude  obtained  was  0.8038  and  the  smallest  was  0.8003.  Four  of  the  corrections 
obtained  were  less  than  0.S010  and  seven  were  less  than  0.8015.  Where  possible  the  program  of 
observations  was  arranged  to  produce  an  accumulation  of  any  existing  relative  personal  equation. 

In  1908  the  difference  of  longitude  between  the  observatory  of  the  new  University  o'f 
Wasliington  at  Seattle  and  the  old  longitude  station  in  Seattle  was  determined,  using  transit 
micrometers.  Observations  were  made  on  six  nights,  the  observers  changing  stations  after 
each  night's  observations.  The  apparent  relative  personal  equation  determined  by  this  method 
of  observation  amounted  to  only  0.008  second. 

The  above  evidence  justifies  the  present  method  of  longitude  observations  with  transit 
micrometers  without  exchange  of  observers.  The  evidence  is  sufficient  to  justify  the  continua- 
tion of  the  present  method  of  carrying  on  telegraphic  longitude  work  for  geographic  and  geodetic 
purposes,  for  the  personal  equation,  if  present,  is  much  smaller  than  the  probable  errors  of  the 
determinations.  However,  where  the  greatest  accuracy  is  required,  as  in  the  determination 
of  the  difference  of  longitude  between  two  fixed  observatories,  then  an  exchange  of  observers 
is  desirable  to  eliminate  any  possible  personal  equation.  An  exchange  of  instruments  is  also 
required  to  eliminate  differences  in  the  total  relay  and  armature  times  at  the  two  ends  of  the 
line.  For  a  complete  elimination  of  this  error  the  adjustments  of  the  relays  and  magnets 
should  be  the  same  before  and  after  exchange. 

The  accuracy  of  the  telegraphic  determination  of  the  difference  of  longitude,  where  no 
transit  micrometer  is  used,  depends  largely  upon  the  accuracy  of  the  determination  of  the  relative 
personal  equation  of  the  two  observers,  and  upon  its  constancy. 

The  relative  personal  equation  of  two  observers  may  be  determined  in  various  ways.  The 
method  to  be  selected  in  a  given  case  depends  upon  circumstances,  involving  the  question  of 
cost,  the  difficulty  of  exchange  of  observers,  and  to  some  degree  the  desired  accuracy  of  the  result. 

In  primary  longitude  determinations,  where  cost  and  ease  of  transportation  are  not  prohibi- 
tive, the  relative  personal  equation  of  the  observers  is  eliminated  from  the  result  by  the  observers 
changing  stations  after  about  one-half  of  the  observing  has  been  done.  In  this  way  the  relative 
personal  equation  will  enter  the  resulting  differences  of  longitude  before  and  after  exchange  of 
observers  with  different  signs  and  the  mean  of  such  determinations  will  be  the  resulting  differ- 
ence of  longitude  with  the  effect  of  personal  equation  eliminated. 

The  relative  personal  equation  may  be  determined  independently  of  the  longitude  observa- 
tions by  the  use  of  two  transits  placed  in  the  same  observatory  or  in  separate  observatories 
close  together,  and  by  having  the  two  observers  observe  independently  the  same  stars,  which 
should  be  arranged  in  time  sets.  If  the  two  instruments  are  on  the  same  meridian,  or  nearly 
so,  and  use  is  made  of  only  one  chronometer  and  chronograph  to  record  both  sets  of  observations, 

1  See  Appendix  2,  Report  for  1897. 


92 


TJ.   S.   COAST   AND  GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 


it  may  be  necessary  to  throw  one  instrument  out  of  adjustment  (in  collimation)  more  than  the 
other  in  order  to  avoid  having  the  observations  overlap.  A  better  arrangement  would  be  to 
have  two  chronographs  controlled  by  the  same  chronometer  by  means  of  local  relays,  and  have 
the  chronograph  records  of  the  two  instruments  independent  of  one  another.  The  difference  of 
the  two  chronometer  corrections  thus  determined,  corrected  for  the  very  small  longitude  differ- 
ence between  the  two  transit  instruments,  is  the  personal  equation  of  the  two  observers.  Some- 
times different  chronometers  are  used  and  compared  in  the  same  manner  as  in  actual  longitude 
determinations. 

The  relative  personal  equation  may  also  be  observed  with  a  single  transit  instrument  as 
follows:  On  the  first  star  A  observes  the  transits  over  the  lines  of  the  first  half  of  the  diaphragm, 
then  quickly  gives  place  to  B  who  observes  the  transits  across  the  remainder  of  the  lines,  omitting 
the  middle  line.  On  the  second  star  B  observes  on  the  first  half  of  the  diaphragm  and  A  follows. 
After  observing  a  series  of  stars  thus,  each  leading  alternately,  each  observer  computes  for  each 
star,  from  the  known  equatorial  intervals  of  the  lines,  and  from  h's  own  observations,  the  time  of 
transit  of  the  star  across  the  mean  line  of  the  diaphragm.  The  difference  of  the  two  deduced  times 
of  transit  across  the  mean  line  is  the  relative  personal  equation.  If  each  has  led  the  same 
number  of  times  in  observing,  the  result  is  independent  of  any  error  in  the  assumed  equatorial 
intervals  of  the  lines.  No  readings  of  the  striding  level  need  be  taken,  and  the  result  is  less 
affected  by  the  instability  of  the  instrument  than  in  the  other  method.  If  the  stars  observed 
by  this  method  are  so  selected  as  to  form  time  sets,  and  the  chronometer  corrections  are  computed 
from  each  observer's  observations  independently,  the  difference  of  these  chronometer  corrections 
will  be  the  relative  personal  equation. 

As  the  accuracy  of  the  telegraphic  determination  of  longitude  without  the  use  of  the  transit 
micrometer  depends  also  upon  the  constancy  of  the  relative  personal  equation  of  the  two  obser- 
vers concerned,  there  is  shown  below  a  table  which  gives  some  values  of  the  relative  personal 
equation  as  derived  from  telegraphic  longitude  observations  (key  and  chronograph  method). 
The  values  in  this  table  indicate  to  what  extent  the  relative  personal  equation  may  be  expected 
to  vary  from  month  to  month  and  year  to  year.  The  plus  sign  indicates  that  the  observer 
first  named  observes  later  (slower)  than  the  other. 

Relative  personal  equation  (not  reduced  to  equator). 


C.  H.  Sinclair—  E.  Smith 
[14  years] 

C.  H.  Sinclair—  R.  A.  Man- 
[4  years] 

C.  H.  Sinclair—  G.  R.  Putnam 
[5  years] 

s           s 

s 

i                                                     s 

s 

1881  Aug.  and  Sept.    -0.123  ±0.008 

1886  Sept.  and  Oct. 

+0.288  ±0 

008 

1891  May  and  June     +0.  184 

±0.011 

1881  Nov.  and  Dec.     -.085           06 

1888  Sept. 

+  .210 

09 

1891  June  and  July     +  .  140 

08 

1885  Apr.  and  May     -  .  047           08 

1888  Oct.  and  Nov. 

+  .144 

11 

1891  July                     +  .172 

06 

1885  May  and  June     -  .  131           03 

1888-9  Dec.  and  Jan. 

.+  .214 

10 

1891  Aug.                     +  .161 

10 

1885  July  and  Aug.     -  .  110           10 

1889  Jan. 

+  .233 

05 

1891  Aug.  and  Sept.    +  .  176 

11 

1886  May  and  June     -  .  062          08 

1889  Jan.  and  Feb. 

+  .225 

07 

1892  Feb.  and  Mar.     +  .  160 

06 

1886  June  and  July     +.010          06 

1889  Feb.  and  Mar. 

+  .267 

07 

1892  Mar.                     +  .  192 

04 

1886  July  and  Aug.     -.023           12 

1889  Mar.  and  Apr. 

+  .278 

12 

1892  Mar.  and  Apr.     +  .  140 

02 

1886  Aug.  and  Sept.    +.056          04 

1889  Apr.  and  May 

+  .217 

12 

1892  Apr.                       +  .  150 

05 

1887  May  and  June     +  .  038           10 

1889  May  and  June 

+  .282 

18 

1892  Apr.  and  May     +  .  126 

04 

1887  June,  July,  and  1 
1+    109           13 

1889  June  and  July 

+  .246 

07 

1892  June  and  July     +  .  109 

10 

Aug.                        JT 

1889  July 

+  .275 

08 

1893  Feb.  and  Mar.     +  .082 

10 

1887  Sept.                    +  .111           13 

1889  July 

+  .265 

05 

1896  Feb.  and  Mar.     +  .  155 

03 

1887  Sept.  and  Oct.     +  .  160          09 

1889  July  and  Aug. 

+  .228 

15 

1896  Mar.                     +  .  129 

07 

1895  Feb.  and  Mar.     +.093           11 

1889  Aug. 

+  .284 

08 

1896  Apr.                     +  .  122 

05 

1895  Mar.                     +  .075           11 

1889  Aug.  and  Sept. 

+  .226 

06 

1896  Apr.  ard  May     +  .  181 

05 

1895  Apr.                     +0.086  ±0.005 

1889  Sept. 

+  .258 

07 

1896  May  and  June     +  .  142 

13 

1890  May  and  June 

+  .  166 

14 

1896  June  and  July     +0.  124 

±0.008 

The  relative  personal  equation  of 

1890  July 

+  .238 

10 

these  two  observers  seems  to  be  a 

1890  July  and  Aug. 

+  .237 

14 

Mean  S.—  P.=              +0.  147 

function  of  the  time  and  a  mean  of 

1890  Aug. 

+0.278  ±0.006 

Prob.  error*  of  a  single  value 

±0.020 

the  above   values   would   therefore 

have  but  little  meaning. 

Mean  S.-M.=' 

+  0.241 

Prob.  error*  of  a  single  value    ±0. 

026 

*  This  value  may  be  taken  as  a  measure  of  the  variability  of  the  personal  equation. 


DETERMINATION   OF   LONGITUDE.  93 

Each  value  in  the  table  depends  upon  8  or  10  nights  of  observation,  4  or  5  nights  each  before 
and  after  the  exchange  of  observers,  and  may  therefore  be  considered  to  be  a  mean  value  covering 
a  period  of  from  two  weeks  to  a  month  or  more.  It  is  improbable  that  the  variation  of  the  rela- 
tive personal  equation  from  night  to  night  is  as  small  as  would  be  inferred  directly  from  the  above 
table.  The  error  due  to  personal  equation,  remaining  in  the  deduced  longitude  after  the 
exchange  of  observers,  is  one-half  the  difference  between  the  mean  value  of  the  relative  personal 
equation  before  the  exchange  of  observers  and  its  mean  value  after  the  exchange. 

DISCUSSION   OF   ERRORS  WHEN   KEY  AND  CHRONOGRAPH  ARE   USED. 

This  discussion  is  based  upon  the  supposition  that  the  regular  program  for  longitude  obser- 
vations when  using  an  observing  key  and  chronograph,  consisting  of  5  nights  each  before  and 
after  exchange  of  observers,  has  been  carried  out,  and  also  that  the  method  of  selection  of  stars 
is  the  one  formerly  in  use  on  primary  longitude  work  in  this  Survey,  in  which  a  time  set  con- 
sisted of  10  stars,  5  before  and  5  after  reversal  of  the  horizontal  axis. 

These  sources  of  error  are  given  the  same  order  as  those  shown  on  pages  85-87  under  the 
heading :  Discussion  of  Errors  when  Transit  Micrometer  is  Used. 

First.  An  accidental  error  arising  from  the  accidental  errors  of  observations  of  200  stars 
at  each  station.  If  the  accidental  error  of  observation  of  a  single  star  be  estimated  at  ±0.810, 
and  this  is  surely  a  sufficiently  large  estimate  to  cover  both  the  observer's  errors  and  those 
instrumental  errors  which  belong  to  the  accidental  class,  the  probable  error  of  the  final  result 
arising  from  this  cause  would  be  ±0.810-^  -JlOQ=  ±0.8010. 

Second.  The  statement  on  page  86  regarding  the  accidental  error  arising  from  the  acci- 
dental errors  in  the  adopted  right  ascensions  of  the  stars  used,  is  applicable  to  all  methods  of 
observing. 

Third.  For  a  statement  regarding  the  errors  due  to  the  variation  of  the  rate  of  the  chrono- 
meter see  page  86. 

Fourth.  Errors  arising  from  the  variation  of  the  relative  personal  equation  from  night  to 
night.  These  are  probably  among  the  largest  errors  involved  in  longitude  determinations.  A 
constant  error,  not  eliminated  by  the  exchange  of  observers,  may  possibly  arise  from  this  source 
if  the  temperature,  altitude,  moisture  conditions,  etc.,  are  very  different  at  the  two  stations. 
Other  than  this,  the  errors  arising  from  this  source  belong  to  the  accidental  class  when  con- 
sidered with  reference  to  the  computed  difference  of  longitude  and  are  exhibited  in  the  residuals 
corresponding  to  the  separate  nights  of  observation. 

Fifth.  The  statement  concerning  errors  due  to  lateral  refraction  on  page  86  is  equally 
applicable  here. 

Sixth.  No  change  is  necessary  in  the  statement  on  page  86  regarding  the  errors  due  to 
variation  in  the  transmission  time. 

Seventh.  The  difference  of  the  transmission  time  through  the  two  signal  relays  enters  as 
an  error  in  the  final  result.  This  error  is  made  very  small  in  the  present  work  of  the  Survey 
by  the  use  of  fast-acting  signal  relays  which  are  as  nearly  alike  as  possible.  It  might  be  further 
reduced  if  each  observer  carried  his  own  switchboard  with  him  when  exchange  of  stations  is  made. 

As  stated  on  page  87,  if  the  difference  in  longitude  which  is  being  measured  is  large,  say 
more  than  30  minutes  of  time,  it  is  well  to  abandon  the  practice  of  endeavoring  to  observe  the 
same  stars  at  both  stations  to  such  an  extent  as  will  bring  the  exchange  of  time  signals  near  the 
middle  of  the  time  observations  at  each  station.  The  error  of  right  ascension  thus  introduced 
will  be  more  than  offset  by  the  accuracy  gained  by  the  proper  placing  of  the  exchange. 

Are  there  appreciable  errors  which  are  constant  for  the  night  in  the  time  determinations 
or  in  the  other  operations  involved  in  the  determination  of  a  longitude  difference  by  the  tele- 
graphic method;  and  if  so,  what  is  the  average  magnitude  of  such  errors?  The  excess  of  the 
probable  error  of  a  longitude  difference  computed  as  indicated  on  page  89  over  its  value  as  de- 
rived from  the  computed  probable  errors  of  the  chronometer  corrections  at  exchange  is  due  to 
errors  which  are  constant  for  and  peculiar  to  each  night.  Using  this  principle l  the  error  peculiar 

1  For  the  formulae  used  in  applying  a  similar  principle  to  latitude  observations,  see  pp.  119-123. 


94  U.   S.   COAST   AND   GEODETIC   SUKVEY   SPECIAL   PUBLICATION    NO.   14. 

to  a  night  has  been  computed  from  fifteen  longitude  determinations  made  since  1890.  It  was 
found  that  the  error  peculiar  to  each  night,  and  therefore  not  capable  of  elimination  by  increasing 
the  number  of  observations  per  night,  expressed  as  a  probable  error,  was  ±  0.S022,  while 
the  probable  error  in  the  result  for  a  night  arising  from  accidental  errors  of  observation,  and 
therefore  capable  of  further  elimination  by  increased  observation,  was  ±  0.S013.  It  should 
be  noted  that  the  errors  discussed  under  all  but  the  first  heading  above  are  each  capable  of  con- 
tributing to  the  error  peculiar  to  a  night.  It  is  likely  that  variation  in  the  personal  equation  is 
the  most  potent  cause  of  such  errors.  It  is  evident  from  the  probable  errors  given  above  that 
very  little  is  lost  in  ultimate  accuracy  if  clouds  interfere  so  as  to  cut  off  a  part,  say  one-fourth, 
of  the  regular  program  of  time  observations  (two  sets  of  ten  stars  each),  and  that  almost  no 
gain  in  accuracy  would  result  from  lengthening  the  program. 

Are  there  appreciable  errors  hi  a  telegraphic  determination  of  a  difference  of  longitude 
which  are  constant  for  the  interval  of  several  days  over  which  the  determination  extends;  and, 
if  so,  what  is  the  average  magnitude  of  such  errors  ?  We  may  obtain  an  answer  to  tliis  question 
by  comparing  the  probable  errors  of  longitude  difference  computed  as  on  page  89  with  the 
same  probable  errors  as  computed  from  the  residuals  developed  in  adjusting  such  a  longitude 
net  as  that  given  in  Appendix  No.  2  of  the  Report  for  1897.  The  excess  of  the  last-named 
probable  errors  over  the  first-named  is  due  to  errors  which  are  constant  for  the  station  during 
the  time  of  occupation.  From  the  published  adjustment  of  the  great  longitude  net  referred 
to  above  (see  pp.  246,  247,  255,  of  Report  for  1897),  after  omitting  the  first  eleven  determinations 
(all  made  not  later  than  1872,  and  several  involving  trans- Atlantic  cables)  and  the  fifty-eighth  de- 
termination (publication  incomplete),  it  follows  that  the  constant  error  peculiar  to  each  longi- 
tude determination  and  not  capable  of  elimination  by  increasing  the  number  of  nights  per  station, 
expressed  as  a  probable  error,  is  ±0."022,  while  the  accidental  error  of  the  deduced  difference 
of  longitude,  which  is  capable  of  further  reduction  by  increasing  the  number  of  nights  per 
station  (beyond  the  standard  number,  ten),  is  ±  0.S011.  It  follows  that  a  reduction  of  the 
number  of  nights  per  station  to  six,  or  even  four,  would  result  in  but  a  slight  decrease  in  accu- 
racy— about  10  per  cent.  Three  sources  of  errors  peculiar  to  a  station  in  the  order  of  their 
probable  magnitude  are  those  mentioned  under  the  fourth,  sixth,  seventh,  and  fifth  headings 
above,  namely:  Variation  in  personal  equation,  variation  in  transmission  time  (especially  when 
a  repeater  interrupts  a  circuit),  the  difference  of  the  two  signal  relay  times,  and  possibly  lateral 
refraction  in  some  cases. 

REDUCTION  TO  MEAN  POSITION  OF  POLE. 

This  correction  will  be  applied  in  the  office  in  accordance  with  the  Preliminary  Results 
published  annually  by  the  International  Geodetic  Association  (see  p.  85). 

A  STATEMENT  OF  COSTS. 

Since  1906  forty-two  differences  in  longitude  have  been  determined  in  the  United  States, 
using  the  transit  micrometer.  Forty-one  were  determined  in  four  seasons.  The  average  cost 
for  the  field  work  and  preparing  for  the  field,  including  all  expenses  and  salaries,  was  $440. 
The  average  cost  per  difference  for  the  various  seasons  varied  from  $360  to  $550.  The  cost  of  a 
difference  of  longitude  between  two  places  will  vary  according  to  the  conditions  under  wluch 
work  is  done,  and  consequently  it  should  be  planned  to  have  the  parties  in  the  field  when  the 
weather  may  be  expected  to  be  most  favorable.  The  work  should  be  localized  for  any  season 
as  much  as  is  possible.  The  longer  the  season  the  more  economically  should  the  work  be  done. 
If  possible,  the  stations  should  be  located  near  the  line  of  the  telegraph  in  order  to  avoid  the 
delay  and  the  expense  of  building  a  long  line  to  the  observatory.  The  determination  of  longi- 
tude differences  telegraphically  in  remote  regions,  such  as  Alaska,  may  cost  from  three  to  six 
or  more  times  the  average  cost  of  a  difference  in  the  United  States. 

No  data  are  readily  available  showing  the  cost  of  the  determination  of  longitudes 
telegraphically,  using  the  key  and  chronograph.  But  owing  to  the  necessity  of  exchanging 


DETERMINATION   OF   LONGITUDE.  95 

observers  for  each  difference  of  longitude  and  of  observing  over  more  nights  than  when  the 
transit  micrometer  is  used,  it  is  probable  that  the  cost  would  be  from  25  to  50  per  cent  more 
than  the  costs  stated  above. 

LONGITUDE   BY  THE  CHRONOMETRIC  METHOD. 

The  equipment,  program  of  observations,  and  methods  of  computation  pertaining  to  a 
determination  of  a  difference  of  longitude  by  the  chronometric  method,  in  which  chronometers 
transported  back  and  forth  between  stations  take  the  place  of  the  telegraphic  signals,  may  be 
most  conveniently  explained  by  giving  a  concrete  example. 

The  longitude  of  a  station  at  Anchorage  Point,  Chilkat  Inlet,  Alaska,  was  determined  in 
1894  by  transporting  chronometers  between  that  station  and  Sitka,  of  which  the  longitude  had 
previously  been  determined.  At  Anchorage  Point  observations  were  taken  on  every  possible 
night  from  May  15  to  August  12,  namely  on  fifty-three  nights,  by  the  eye  and  ear  method, 
using  a  meridan  telescope.  The  hack  or  observing  chronometer  kept  sidereal  time,  and  there 
were  also  four  other  chronometers  at  the  station,  two  keeping  mean  time  and  two  sidereal.  These 
four  chronometers  were  never  removed  during  the  season  from  the  padded  double-walled  box  in 
which  they  were  kept  for  protection  against  sudden  changes  of  temperature  and  in  which  the 
hack  chronometer  was  also  kept  when  not  in  use.  The  instrumental  equipment  and  procedure 
at  Sitka  was  similar  to  that  just  described.  A  sidereal  chronometer  was  the  hack,  and  two  other 
chronometers,  one  sidereal  and  one  mean  time,  were  used  in  addition.  Nine  chronometers,  eight 
keeping  mean  tune  and  one  sidereal,  were  carried  back  and  forth  between  the  stations  on  the 
steamer  Hassler. 

Aside  from  the  time  observations,  the  programme  of  operations  was  as  follows :  Just  before 
beginning  the  time  observations  at  Anchorage  Point,  and  again  as  soon  as  they  were  finished  on 
each  night,  the  hack  chronometer  was  compared  with  the  two  mean  time  chronometers  by  the 
method  of  coincidence  of  beats  (described  on  p.  96).  These  two  were  then  compared  with 
each  of  the  two  remaining  (sidereal)  chronometers  at  the  station.  These  comparisons,  together 
with  the  transit  time  observations,  served  to  determine  the  correction  of  each  chronometer  to 
local  time  at  the  epoch  of  the  transit  observations.  Whenever  the  steamer  first  arrived  at  the 
station,  and  again  when  it  was  about  to  leave,  the  hack  chronometer  was  compared  with  the 
other  station  chronometers,  as  indicated  above,  was  carried  on  board  the  steamer  and  compared 
with  the  nine  traveling  chronometers,  and  then  immediately  returned  to  the  station  and  again 
compared  with  the  other  four  station  chronometers.  On  board  the  steamer  the  hack  was  com- 
pared by  coincidence  of  beats  with  each  of  the  eight  mean  time  chronometers,  and  the  remaining 
(sidereal)  chronometer  was  then  compared  with  some  of  the  eight.  The  comparisons  on  shore 
before  and  after  the  trip  to  the  steamer  served  to  determine  the  correction  of  the  hack  at  the 
epoch  of  the  steamer  comparisons.  The  steamer  comparisons  *  determined  the  corrections  of 
each  of  the  traveling  chronometers  to  Anchorage  Point  time.  Similar  operations  at  Sitka  deter- 
mined the  corrections  of  the  nine  traveling  chronometers  to  Sitka  time  as  soon  as  they  arrived 
and  again  just  before  they  departed  from  Sitka.  During  the  season  the  steamer  made  seven 
and  a  half  round  trips  between  the  stations. 

CARE  OF  CHRONOMETERS. 

To  secure  the  greatest  possible  uniformity  of  rate  a  chronometer  should  be  kept  running 
continuously,  both  when  in  use  and  when  out  of  use  between  seasons  of  work.  When  it  is 
allowed  to  remain  stopped  for  a  considerable  time,  the  oil  in  the  bearings  tends  to  become  gummy. 
When  started  again,  the  chronometer  will  tend  to  have  a  varying  rate  for  some  time  until  the 
effects  of  the  stoppage  have  been  worn  off. 

If  a  chronometer  is  to  be  shipped  (by  express,  for  example),  and  therefore  is  to  be  subjected 
presumably  to  comparatively  violent  handling  and  jarring,  it  should  always  be  stopped  and  the 
balance  wheel  locked  by  gently  inserting  small  wedge-shaped  pieces  of  clean  cork  under  it. 

1  In  addition  to  the  chronometer  comparisons  referred  to  in  this  paragraph  the  steamer  chronometers  and  the  station  chronometers  were  each 
intercompared  daily.    This  was  done  merely  as  a  check  upon  their  performance. 


96  U.   S.   COAST   AND  GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.    14. 

A  running  chronometer  should  always  be  protected  as  carefully  as  possible  against  jars, 
and  especially  against  such  sharp  quick  jars  as  result  from  setting  it  down  upon  a  hard  surface. 
Either  the  surface  upon  which  it  is  set  should  be  padded  or  a  cushion  should  be  carried  with  the 
chronometer.  When  it  becomes  necessary  to  carry  a  chronometer  in  the  hand — as,  for  example, 
when  a  hack  chronometer  is  carried  back  and  forth  between  an  observatory  and  a  steamer  in  con- 
nection with  chronometric  longitudes — the  gimbals  should  be  locked  to  prevent  the  chronometer 
from  swinging.  It  is  important  that  the  locking  should  be  done  in  such  a  way  that  there  will  be 
no  looseness  and  the  corresponding  tendency  to  a  chucking  motion.  While  the  chronometer  is 
being  carried,  swinging  of  the  arm  should  be  avoided  as  much  as  possible.  Any  swinging  of 
the  chronometer  in  azimuth  is  especially  objectionable,  as  it  tends  to  make  it  skip  seconds  and 
to  damage  it.  Chronometers  have  been  known  to  skip  seconds,  probably  from  this  cause,  even 
in  the  hands  of  an  experienced  and  careful  officer.  On  shipboard  chronometers  should  be  left 
free  to  swing  in  their  gimbals,  which  should  be  so  adjusted  that  the  face  of  the  chronometer  will 
be  approximately  horizontal.  Any  change  in  this  adjustment  is  apt  to  produce  a  change  of  rate. 

COMPARISON  OF  CHRONOMETERS  BY  COINCIDENCE  OF  BEATS. 

The  process  of  comparing  a  sidereal  and  a  mean  time  chronometer  is  analogous  to  that  of 
reading  a  vernier.  The  sidereal  chronometer  gains  gradually  on  the  mean  time  chronometer, 
and  once  in  about  three  minutes  the  two  chronometers  tick  exactly  together  (one  beat  =  0". 5). 
As  one  looks  along  a  vernier  to  find  a  coincidence,  so  one  listens  to  this  audible  vernier  and  waits 
for  a  coincidence.  As  in  reading  a  vernier  one  should  look  at  lines  on  each  side  of  the  supposed 
coincidence  to  check,  and  perhaps  correct  the  reading  by  observing  the  symmetry  of  adjacent 
lines,  so  here  one  listens  for  the  approaching  coincidence,  hears  the  ticks  nearly  together,  appar- 
ently hears  them  exactly  together  for  a  few  seconds,  and  then  hears  them  begin  to  separate, 
and  notes  the  real  coincidence  as  being  at  the  instant  of  symmetry.  The  time  of  coincidence  is 
noted  by  the  face  of  one  of  the  chronometers.  Just  before  or  just  after  the  observation  of  the 
coincidence  the  difference  of  the  seconds  readings  of  the  two  chronometers  is  noted  to  the  nearest 
half  second  (either  mentally  or  on  paper).  This  difference  serves  to  give  the  seconds  reading 
of  the  second  chronometer  at  the  instant  of  coincidence.  The  hours  and  minutes  of  both  chro- 
nometers are  observed  directly.  When  a  number  of  chronometers  are  to  be  intercompared,  the 
experienced  observer  is  able  to  pick  out  from  among  them  two  that  are  about  to  coincide.  He 
compares  those,  selects  two  more  that  are  about  to  coincide  and  compares  them,  and  so  on; 
and  thus  to  a  certain  extent  avoids  the  waits,  of  a  minute  and  a  half  on  an  average,  which  would 
otherwise  be  necessary  to  secure  an  observation  on  a  pair  of  chronometers  selected  arbitrarily. 

At  Sitka  on  July  13,  1894,  it  was  observed  that  18h  30m  088.00  on  chronometer  No.  194 
(sidereal)  =  llh  52m  308.00  on  chronometer  No.  208  (mean  time);  and  that  llh  15m  35s. 50  on 
chronometer  No.  1510  (mean  time)  =  14h  48m  108.00  on  chronometer  No.  387  (sidereal).  It 
was  known  that  at  the  epoch  of  the  comparisons  the  correction  of  No.  194  to  Sitka  sidereal 
time  was  -lm  548.01,  and  of  No.  1510  to  Sitka  mean  tune  was  -6m  268.34.  The  required 
corrections  to  No.  208  and  No.  387  were  computed  as  follows: 

ft       nt         »  A       m         « 

Time  by  194  =18    30    08. 00  Time  by  1510  =  11    15    35. 50 

Correction  to  194  =     -01    54. 01  Correction  to  1510  =       -  6    26. 34 


Sidereal  time  =18    28    13. 99  Mean  time  =  11    09    09. 16 

Sidereal  time  of  mean  noon=  7    26    53. 66  Correction  mean  to  sidereal    =       +01    49.  93 


Sidereal  interval          =11    01    20. 33  Sidereal  interval  =  11    10    59. 09 

Correction,  sidereal  to  mean  =     —01    48.  34  Sidereal  time  of  mean  noon=     7    26    53.  66 


Mean  time  =10    59    31. 99  Sidereal  time  =  18    37    52. 75 

Time  by  208  =11    52    30. 00  Time  by  387  =  14    48    10. 00 


Correction  to  208          =     -52    28.01  Correction  to  387          =+3    49    42.75 

The  correction  to  reduce  a  sidereal  to  a  mean  time  interval,  or  vice  versa,  may  be  taken 
from  the  tables  in  the  back  part  of  the  American  Ephemeris.     The  sidereal  time  of  mean  noon 


DETERMINATION   OF    LONGITUDE. 


97 


may  be  taken  from  that  part  of  the  Ephemeris  headed  "Solar  ephemera,"  and  it  should  not  be 
overlooked  that  it  is  the  sidereal  time  of  local  mean  noon  that  is  required,  and  that,  therefore,  the 
longitude  (approximate)  of  the  station  must  be  taken  into  account.  The  correction  to  be 
applied  to  Washington  sidereal  time  of  mean  noon  to  obtain  that  for  the  station  is  the  same  as 
the  correction  to  reduce  a  mean  time  interval  equal  to  the  longitude  of  the  station  from  Wash- 
ington to  a  sidereal  interval. 

COMPUTATION  OF  LONGITUDE  FROM  A  SINGLE  ROUND  TRIP. 

From  the  operations  at  Anchorage  Point  the  correction  of  each  station  chronometer  at  the 
epoch  of  each  set  of  time  observations  became  known.  The  intercomparisons  on  shore  before 
leaving  for  the  steamer  and  after  returning,  together  with  the  assumption  that  each  station 
chronometer  runs  at  a  uniform  rate  between  time  sets,  gave  five  separate  determinations  of  the 
correction  to  the  hack  at  the  epoch  of  the  steamer  comparisons. 

Thus,  on  June  18,  1894,  at  3h.45  by  its  own  face,  the  middle  epoch  of  the  steamer  com- 
parisons, the  correction  to  the  hack  (No.  380)  was 


By  its  own  rate       -2    38. 16  (weight 


By  No.  4969  rated 
By  No.  2490  rated 
By  No.  207  rated 
By  No.  2637  rated 


38.30 
38.26 
38.16 

38.  62  (weight  f). 


Mean         =  -2    38.  30 
Weighted  mean  =-2    38.25 

The  comparisons  of  No.  380  with  No.  4969  at  the  station  on  this  date,  computed  upon  the 
supposition  that  No.  4969  ran  at  a  uniform  rate  between  preceding  and  following  time  observa- 
tions, showed  that  the  correction  to  No.  380  at  2h.64  by  its  face  was  -2m  38S.34,  and  at  4h.36 
was  —  2m  38S.25.  Assuming  it  to  run  uniformly  between  these  epochs,  its  correction  was  —  2m 
388.30  at  3h.45,  as  shown  above. 

An  examination  of  the  daily  rates  of  the  five  chronometers  showed  that  No.  2637  ran  very 
irregularly,  and  that  No.  380  did  not  run  as  regularly  as  the  other  three.  Hence  these  chro- 
nometers were  assigned  less  weight  than  the  others,  as  indicated  above.1 

Using  the  weighted  mean  value  for  the  correction  to  No.  380  at  the  epoch  of  the  steamer 
comparisons  these  comparisons  give  the  correction  of  each  traveling  chronometer  on  Anchorage 
Point  time. 

Similar  operations  at  Sitka  gave  the  correction  to  each  traveling  chronometer  on  Sitka 
tune  on  each  arrival  at  and  departure  from  Sitka. 

Computation  of  difference  of  longitude  of  Sitka  and  Anchorage  Point. 

FIRST   TRIP  STARTING   FROM   ANCHORAGE    POINT. 


Chronomc- 

Anchorage  Point. 
May  15 

Sitka,  May  17 

Sitka,  May  20 

Anchorage  Point, 
May  23 

M.  T.  or  SU. 

Chr. 
epoch 

Correction 

Chr. 
epoch 

Correction 

Chr. 

epoch 

Correction 

ep«h         Correction 

h 

h   m      s 

h 

h  m      s 

h 

h  m      s 

h 

h  m      « 

M.  T.     231 

11.83 

-0  03  31.39 

7.54 

-0  03  02.  93 

7.55 

-0  03  02.  14 

7.65 

—0  03  20.  -'6 

1  607 

11.84 

-0  01  03.88 

7.81 

-0  CO  34.  93 

7.67 

-0  00  33.  73 

7.65 

-0  01  01.34 

1  510 

12.15 

-0  03  42.50 

7.75 

-0  03  19.  43 

7.52 

-003  28.22 

7.75 

-0  04  05.  90 

196 

9.49 

+2  26  28.51 

5.20 

+2  2653.00 

5.19 

+2  26  46.08 

5.29 

+2  26  1C.  72 

1  542 

11.92 

-0  02  55.  84 

7.53 

-0  02  29.  37 

7.72 

-002  31.83 

7.81 

-0  03  02.  63 

1  728 

9.38 

+2  34  40.  23 

5.08 

+2  34  59.90 

4.91 

+2  34  46.00 

5.23 

+2  34  02.  4(j 

208 

12.71 

-0  42  08.  24 

8.17 

-0  42  01.19 

8.48 

-0  42  35.76 

8.56 

-0  43  35.37 

2  167 

8.73 

+3  18  39.  99 

4.39 

+3  19  09.98 

4.15 

+3  19  12.69 

4.59 

+3  18  47.44 

Sid.     387 

11.97 

+3  46  50.04 

7.65 

+3  47  22.97 

7.7S          +3  47  29.07 

8.29         +3  47  09.31 

i  If  considered  desirable,  the  relative  weights  to  be  assigned  to  the  station  chronometers  may  be  determined  more  accurately  by  the  method 
outlined  in  the  footnote  on  p.  100. 

8136°— 13 7 


98 


U.   S.   COAST   AND  GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.   14. 

Computation  of  difference  of  longitude  of  Sifka  and  Anchorage  Point — Continued. 

FIRST  TRII'  STARTING  FROM  ANCHORAGE  POINT— Continued. 


From  Anchor- 

Chro- 
nometers 

Total 

At  Sitka 

Traveling 

Dailv 

age  Point  to 
Sitka 

Correction 
at  Sitka 
on  Anchor- 

Differ- 
ence of 

M.  T.or 
Sid. 

rate 

age  Point 

longi- 
tude 

Time 

Rate 

Time 

Rate 

Time 

Rate 

Time 

Rat« 

d     h 

s 

d     h 

s 

d     h 

s 

s 

d     h 

s 

h    m      s 

m      s 

M.T.231 

7  19.82 

+  1.93 

2  24.01 

+  0.79 

4  19.81 

+  1.14 

+  0.24 

1  19.71 

+  0.43 

-0  03  30.  96 

0  28.  03 

1  507 

.81 

+  2.54 

23.86 

+  1.20 

19.95 

+  1.34 

+  0.28 

19.97 

+  0.51 

-0  01  03.37 

28.44 

1  510 

.60 

-23.40 

23.77 

-  8.79 

19.83 

-14.61 

-  3.03 

19.60 

-  5.50 

-0  03  48.00 

28.57 

196 

.80 

-17.79 

23.99 

-  6.92 

19.81 

-10.  87 

-2.25 

19.71 

-  4.10 

+2  26  24.41 

28.59 

1  542 

.89 

-  6.79 

24.19 

-  2.46 

19.70 

-  4.33 

-0.90 

19.61 

-1.64 

-0  02  57.48 

28.11 

1  728 

.85 

-37.77 

23.83 

-13.90 

20.02 

-23.  87 

-  4.94 

19.70 

-  8.99 

+2  34  31.24 

28.66 

208 

.85 

-90.13 

24.31 

-34.57 

19.54 

-55.56 

-11.54 

19.46 

-20.90 

-0  42  29.  14 

27.95 

2  167 

.86 

+  7.45 

23.76 

+  2.71 

20.10 

+  4.74 

+  0.98 

19.66 

+  1.78 

+3  18  41.77 

28.21 

Sid.  387 

20.32 

+19.27 

24.13 

+  6.70 

20.19 

+12.57 

+  2.60 

19.68 

+  4.73 

+3  46  54.  77 

28.20 

In  the  form  on  page  97  the  column  headed  "Chr.  epoch"  gives  the  face  reading  of  the  chro- 
nometer, expressed  in  hours  and  hundredths  (rather  than  minutes  and  seconds)  for  convenience 
in  computation.  The  corrections  at  Anchorage  Point  are  to  the  local  time  of  that  station,  and 
at  Sitka  to  Sitka  local  time. 

In  the  form  above,  the  second  and  third  columns  give  the  elapsed  chronometer  time  and 
the  accumulated  rate  between  the  Anchorage  Point  steamer  comparisons,  and  the  fourth  and 
fifth  columns  give  the  same  quantities  between  the  Sitka  steamer  comparisons.  The  second 
column  minus  the  fourth,  and  the  third  minus  the  fifth  are  the  traveling  time  (both  ways)  and 
the  accumulated  rate  while  traveling,  from  which  the  daily  traveling  rate  as  given  in  the  eighth 
column  becomes  known.  The  ninth  column  gives  the  traveling  time  between  steamer  com- 
parisons from  Anchorage  Point  to  Sitka,  and  the  tenth  column  gives  the  accumulated  rate  dur- 
ing this  interval  computed  by  the  use  of  the  eighth  column.  This  accumulated  rate  being 
applied  as  a  correction  to  the  chronometer  correction  on  Anchorage  Point  time  at  the  begin- 
ning of  the  trip  gives  the  correction  on  Anchorage  Point  time  on  arrival  at  Sitka.  This 
difference  subtracted  from  the  directly  observed  correction  on  Sitka  time  at  that  epoch,  shown 
in  the  upper  form,  gives  the  required  difference  of  longitude. 

It  should  be  noted  that  in  this  computation  the  traveling  rate  is  supposed  to  be  a  constant 
during  the  round  trip,  but  is  not  assumed  to  be  the  same  as  the  rate  while  in  port. 

The  longitude  difference  if  computed  from  the  return  half  of  the  trip,  from  Sitka  to  Anchor- 
age Pjint,  would  necessarily  by  this  process  of  computation  be  identical  with  that  shown  above. 

If  the  steamer  had  stopped  so  short  a  time  at  Sitka  that  only  one  set  of  steamer  compari- 
sons had  been  made  while  there,  as  was  frequently  the  case,  the  above  computation  would  have 
been  simplified  in  an  obvious  manner. 

COMBINATION  OF  RESULTS. 

The  difference  of  longitude  was  thus  computed  from  each  traveling  chronometer  for 
each  round  trip,  starting  from  Anchorage  Point,  the  last  half  trip  (iy2  round  trips  being  made) 
from  Anchorage  Point  to  Sitka,  being  omitted.  A  similar  computation  was  also  made  for 
each  round  trip,  starting  from  Sitka,  the  first  half  trip,  Anchorage  Point  to  Sitka,  now  being 
omitted.1  Each  of  these  computations  would  be  subject  to  a  constant  error  if  the  traveling 
chronometers  had  uniformly  accelerated  or  uniformly  retarded  rates,  but  their  mean  is  free 
from  this  error.  One  half  of  the  computation  also  serves  as  a  check  on  the  other  half. 

i  If  the  steamer  had  returned  again  to  Anchorage  Point,  so  as  to  complete  eight  round  trips,  all  of  the  eight  would  have  been  used  in  the 
first  computation;  and  in  the  second  computation  (round  trips,  starting  from  Sitka)  the  last  trip  from  Sitka  to  Anchorage  Point,  combined  with 
ihe  first  trip  in  the  opposite  direction,  would  have  been  used  as  the  eighth  round  trip.  This  principle  of  computing  the  difference  of  longitude 
from  the  round  trips  starting  from  each  station  in  turn,  and  combining  the  two  results  was  used  for  the  first  time  by  Assistant  C.  A.  Schott  in 
1857  in  deriving  the  difference  of  longitude  of  Savannah,  Ga.,  and  Fernandina,  Fla.  (See  Coast  Survey  Report  for  1857,  pp.  314-324.) 


DETERMINATION    OF   LONGITUDE. 

The  method  of  combining  these  separate  results  is  shown  in  the  following  form . 
Difference  of  longitude  between  Siika  and  Anchorage  Point,  ChilJcat  Inlet,  Alaska. 

SUMMARY  OF  RESULTS  FROM   SEVEN  ROUND  TRIPS,  STARTING  FROM  ANCHORAGE  POINT. 


99 


Chronometers, 
M.  T.  or  Sid. 

l.t                2d                3d               4th               5th               6lh               7lh 

Means 
JA 

Weights 

S                   S                   S                   S                 S                   S                   S 

S 

M.  T.              231 

28.  03      26.  36      28.  36      28.  19      28.  45      28.  19      28.  18 

27.97 

3 

1507 

28.  44      29.  06      29.  18      28.  26      28.  27      28.  20      28.  54 

28.56 

4 

1510 

28.57      29.25      29.00      28.52      28.63      28.06      28.58 

28.66 

7 

196 

28.59      29.09      29.54      28.59      28.43      28.51      28.92 

28.81 

3 

1542 

28.  11      28.  11      28.  66      28.  23      28.  47      28.  38      28.  37 

28.33 

22 

1728 

28.  66      28.  94      29.  16      28.  63      28.  58      28.  43      28.  59 

28.71 

6 

208 

27.  95      27.  40      28.  21      28.  19      28.  42      28.  42      28.  09 

28.10 

6 

2167 

28.21      28.56      28.90      28.55      28.68      28.27      28.64 

28.54 

17 

Sid.              387 

28.20      28.44      28.91      27.93      28.41       27.93      28.59 

28.34 

6 

Mean 

28.31      28.36      28.88      28.34      28.48      28.27      28.50 

28.45 

Weighted  mean 

28.25      28.38      28.82      28.35      28.52      28.28      28.49 

28.44 

Weight 

3122212 

Weighted  mean  0"    Om    28'.44±0S.05 

SUMMARY  OF  RESULTS  FROM  SEVEN  ROUND  TRIPS,  STARTING  FROM  SITKA. 


Chronometers, 
M.  T.  or  Sid. 

l»t                2<i                3d               4'h               5«>>               6th               7"> 

Means 

a 

Weights 

S                  S                   S                  S                   S                  S                   S 

S 

M.  T.           231 

28.87      28.78      28.74      28.39      28.37      28.71      28.11 

28.57 

3 

1507 

27.  69      29.  08      29.  11       27.  76      28.  78      27.  93       28.  64 

28.43 

4 

1510 

28.37      28.88      28.82      27.91      28.83      28.10      28.58 

28.50 

7 

196 

28.59      29.07      28.95      27.66      28.03      29.56      29.20 

28.72 

3 

1542 

28.93      28.57      28.59      28.22      28.50      28.50      28.32 

28.52 

22 

1728 

27.59      28.90      28.75      27.99      29.01      28.09      28.75 

28.44 

6 

208 

27.71      28.03      28.52      28.58      27.88      28.76      27.65 

28.16 

6 

2167 

28.24      28.71      28.80      28.27      28.77      28.31      28.49 

28.51 

17 

Sid.              387 

28.  68      28.  80      28.  43      27.  69      28.  97      27.  98      28.  73 

28.47 

6 

Mean 

28.30      28.76      28.75      28.05      28.57      28.44       28.50 

28.48 

Weighted  mean 

28.  41      28.  69      28.  70      28.  13      28.  61      28.  38      28.  44 

28.48 

Weight 

1222222 

Weighted  mean  Ch    0" 


Final  mean  AX 


2SM8±0«.05 

h     m 
=  +0    00 


28.46±0.05 


Let  N  be  the  number  of  days  during  which  the  chronometers  are  depended  upon  to  carry 
the  time  during  each  round  trip,  reckoned  by  adding  to  the  "traveling  time,"  as  given  in  the 
sixth  column  of  the  form  on  page  98,  the  interval  between  each  comparison  of  the  hack  chro- 
nometer with  the  traveling  chronometers  and  the  nearest  (either  before  or  after)  time  obser- 
vation made  at  that  station.  The  weight  assigned  to  each  trip  is  proportional  to  the  reciprocal 
of  N.  This  weighting  depends  upon  the  assumptions  that  errors  in  the  computed  longitude 
arising  from  the  time  determinations  and  from  the  chronometer  comparisons  are  small  as 
compared  with  those  arising  from  variations  in  Chronometer  rates;  that  the  time  is  carried  by 
the  combined  station  chronometers  over  the  intervals  during  which  they  are  depended  upon 
with  about  the  same  degree  of  accuracy  (due  regard  being  paid  to  the  length  of  the  interval) 
as  the  combined  traveling  chronometers  carry  the  time  during  the  trip,  and,  finally,  that  the 
errors  arising  from  the  variations  in  the  chronometer  rates  belong  to  the  accidental  class  and 
are  proportional  to  the  square  root  of  the  length  of  the  interval  over  which  the  time  is  carried. 


100 


U.   S.   COAST   AND   GEODETIC    SUKVEY   SPECIAL   PUBLICATION    NO.   14. 


WEIGHTS  ASSIGNED   TO   SEPARATE   CHRONOMETERS. 

Even  a  cursory  examination  of  such  a  table  as  that  given  on  the  preceding  page  shows 
that  some  chronometers  run  much  more  uniformly  than  others,  and  therefore  furnish  determina- 
tions of  the  longitude  difference  which  are  entitled  to  greater  weight.  Let  Z1;  12, 13, .  .  .  la  be  the 
derived  values  of  the  difference  of  longitude  as  given  by  one  chronometer  on  the  different  trips, 
and  let  I  be  their  mean.  Let  n  be  the  number  of  trips.  Then,  by  the  ordinary  laws  of  least 
squares,  assigning  equal  weights  to  the  separate  trips,  the  probable  error  of  any  one  of  these 
Z'sis 

.    .    q-Q'T 


71-1 

The  weight  p,  inversely  proportional  to  the  square  of  this  probable  error  to  be  assigned  to 
a  chronometer,  is  proportional  to 

71-1 


The  computation  of  weights  may  be  put  in  the  following  convenient  tabular  form: 

COMPUTATION   OF  WEIGHTS. 
From  the  seven  round  trips  starting  from  Anchorage  Point. 


Chronometer 
I 

231 
27«.97 

1507 
2S-.56 

1510 
2S-.66 

196 
28-.81 

1542 
2S-.33 

1728 
28-.71 

208 
28'.  10 

2167 
28-.S4 

337 
2S-.34 

l-l, 
l-l, 
1-13 
1-1, 
l-k 
l-l« 
1-17 

(l-J,)» 
(l-J,)' 
(1-13? 

ft-W 

(l-k)* 
(l-k? 
(l-lrf 

Z(l-W 
By  2d  comb.* 
Mean  of  2 
n-1 
n-1 

-    .06 
+  1.61 
-    .39 

-  .22 
-  .48 
-  .22 

-  .21 

+  .12 
-  .50 
-  .62 
+  .30 
+  .29 
+  .36 
+  .02 

+   .09 
-  .59 
-  .34 
+  -14 
+  .03 
+  .60 
+  .08 

+   .22 
-  .28 
-  .73 
+  .22 
+  .38 
+  .30 
-  .11 

+  .22 
+  .22 
-  .33 
+  .10 
-  .14 
-  .05 
-  .04 

+   .05 
-  .23 
-  .45 
+  .08 
+  .13 
+  .28 
+  -12 

+   .15 
+  .70 
-  .11 
-  .09 
-  .32 
-  .32 
+  .01 

+  .33 

-    .02 

-  .36 
-  .01 
-  .14 

+  .27 
-  .10 

+  .14 
-  .10 
-  .57 
+  -41 
-  .07 
+  -41 
-  .25 

.00 
2.59 
.15 
.05 
.23 
.05 
.04 

.01 
.25 
.38 
.09 
.08 
.  13 
.00 

.01 
.35 
.  12 
.02 
.00 
.36 
.01 

.05 
.08 
.53 
.05 
.14 
.09 
.01 

.05 
.05 
.11 
.01 
.02 
.00 
.00 

.00 
.05 
.20 
.01 
.02 
.08 
.01 

.02 
.49 
.01 
.01 
.10 
.10 
.00 

.11 
.00 
.  13 
.00 
.02 
.07 
.01 

.02 
.01 
.32 
.  17 
.00 
.17 
.06 

3.11 

.47 
1.79 
6 

3.3 

.94 
2.30 
1.62 
6 

3.7 

.87 
.89 
0.88 
6 

6.8 

.95 

2.73 
1.84 
6 

3.3 

.24 
.30 
0.27 
6 

22.  2 

.37 
1.78 
1.08 
6 

5.6 

.73 
1.22 
0.98 
6 

6.  1 

.34 
.36 
0.35 
6 

17.0 

.75 
1.32 
1.04 
6 

5.8 

2-(l-ln)' 

*  From  similar  results  from  seven  round  trips  starting  from  Sitka. 

A  similar  computation  was  made  using  the  seven  round  trips  starting  from  Sitka,  the  results 
of  which  are  shown  in  the  line  marked  "by  2d  combination,"  and  the  weights  were  derived 
from  the  mean  results  of  the  two  computations.1 

DISCUSSION   OF  ERRORS. 

The  error  in  a  difference  of  longitude  observed  and  computed  as  indicated  in  the  preced- 
ing sections  depends  upon  the  errors  in  the  transit  tune  observations,  errors  in  the  comparison 
of  chronometers,  errors  arising  from  variations  in  the  rates  of  chronometers,  and,  finally,  the 
relative  personal  equation  of  the  two  observers  concerned. 

i  The  relative  weights  to  be  assigned  to  the  station  chronometers  when  they  are  used  to  determine  the  correction  of  the  hack  at  the  epoch  of 
the  steamer  comparisons  might  be  computed  by  an  analogous  process.  Let  O  be  the  correction  to  a  chronometer  at  the  epoch  of  transit  time  obser- 
vations as  determined  from  those  observations.  Let  /  be  its  correction  at  that  same  epoch  interpolated  between  its  observed  corrections  at  the  last 
preceding  and  first  following  transit  time  observations  on  the  assumption  that  its  rate  during  that  interval  is  constant.  For  a  group  of  chronome- 
ters whose  corrections  are  all  determined  a  number  of  times  in  succession  by  the  same  transit  observations,  the  relative  weights  are  evidently 

proportional  to  ^  ij_Q\r 


DETERMINATION   OF   LONGITUDE.  101 

The  errors  in  the  time  observations  will  in  general  be  very  small  in  co.nparison  with  the 
other  errors  affecting  the  result.  For  the  probable  magnitude  of  the  time  errors  see  the  first 
part  of  this  publication.  In  Appendix  No.  3  of  the  Report  for  1894  and  in  No.  3  of  1895  may 
be  found  detailed  statements  of  the  results  of  several  determinations  of  longitude  by  the  chro- 
nometric  method  which  will  serve  to  give  a  concrete  idea  of  the  magnitude  of  the  errors  involved 
in  such  determinations.  The  relative  magnitude  of  the  errors  arising  from  the  time  determi- 
nations increases  as  the  time,  N  (see  p.  99),  required  for  a  round  trip  decreases. 

The  errors  made  in  comparing  chronometers  by  the  method  of  coincidences  are  negligible 
in  then-  effect  upon  the  final  result.  The  checks  obtained  during  the  intercomparisons  of 
chronometers  show  that  the  probable  error  in  a  single  comparison  is  about  ±08.01,  correspond- 
ing to  a  probable  error  of  about  ±48  in  estimating  the  time  of  coincidence  of  ticks. 

The  errors  arising  from  variations  in  the  rates  of  chronometers  are  by  far  the  most  serious 
class  of  errors  involved  in  chronometric  determinations  of  longitude.  The  table  of  results 
given  on  page  99  gives  a  fair  indication  of  the  magnitude  of  the  errors  to  be  expected  from  this 
source. 

The  various  traveling  chronometers  are  subjected  to  variations  of  temperature,  humidity, 
and  barometric  pressure,  and  to  disturbances  arising  from  the  motion  of  the  ship,  which  are 
common  to  them  all.  Do  these  common  conditions  produce  variations  in  rate  which  are  common 
to  all  the  chronometers,  and  therefore  introduce  a  common  error  into  the  various  values  of  the 
longitude  difference  resulting  from  any  one  trip  ?  An  examination  of  the  results  of  six  chrono- 
metric determinations  of  longitude  in  Alaska,  printed  in  the  1894  and  1895  Reports,  indicates 
that  such  errors  in  the  deduced  longitudes,  common  to  all  the  chronometers  on  a  given  trip, 
are  exceedingly  small  upon  an  average — so  small  that  they  are  concealed  by  the  accidental 
errors. 

Chronometers  are  compensated  for  temperature  as  well  as  possible  by  the  maker,  but 
such  compensation  is  necessarily  somewhat  imperfect.  In  general,  however,  this  compensa- 
tion is  so  nearly  perfect  that  little  or  nothing  is  gained  in  accuracy  by  deriving  and  using  tem- 
perature coefficients  connecting  the  temperature  and  the  rate.  There  are  occasional  excep- 
tions; for  example,  the  Button  chronometer  No.  194  (see  pp.  77-78  of  the  Report  for  1894) 
shows  a  very  large  variation  in  rate  due  to  change  of  temperature. 

In  considering  the  errors  due  to  variations  in  chronometer  rates  it  should  not  be  overlooked 
that  the  station  chronometers  are  depended  upon  to  carry  the  time  over  the  interval  from  the 
nearest  time  observations  to  the  steamer  comparisons  in  precisely  the  same  manner  in  which 
the  traveling  chronometers  are  depended  upon  during  the  trip.  It  is  because  of  this  fact  that 
it  may  be  desirable  during  periods  of  very  bad  weather  to  supplement  the  transit  observations 
upon  stars  by  transit  observations  upon  the  sun,  as  indicated  on  page  51,  or  in  low  latitudes  by 
theodolite  or  vertical  circle  observations  for  tune,  or  even  by  sextant  observations  for  time. 

Unless  the  relative  personal  equation  is  eliminated  from  the  computed  longitude  it  is  apt 
to  be  one  of  the  largest  errors  affecting  the  mean  result,  except  when  the  round  trips  are  very 
long  or  very  few  chronometers  are  carried.  It  may  be  eliminated  by  any  of  the  methods  sug- 
gested on  pages  90-93. 

Assuming  that  the  relative  personal  equation  is  eliminated  by  direct  determination  or 
otherwise,  the  error  of  the  mean  result  of  a  chronometric  longitude  determination  will  be  nearly 
inversely  proportional  to  the  square  root  of  the  number  of  chronometers  carried  (provided  the 
stations  are  supplied  with  a  sufficient  number  of  good  chronometers  to  make  the  shore  errors 
small),  to  the  square  root  of  the  number  of  round  trips,  and  the  square  root  of  the  average  value 
of  N  (the  interval  over  which  the  time  is  carried  by  the  chronometers).  It  will  depend  very  inti- 
mately upon  the  quality  of  the  chronometers  and  upon  the  care  with  which  they  are  protected 
from  temperature  changes  and  jars.  It  will  be  affected  very  little  by  an  increase  in  the  errors  of 
the  time  observations  proper,  resulting  from  very  fragmentary  observations  on  cloudy  nights  or 
from  substituting  some  more  approximate  method  for  transit  observations  upon  stars. 

From  the  above  principles  and  the  numerical  values  given  in  Appendix  No.  3  of  the  1894 
Report  and  in  No.  3  of  the  1895  Report,  one  may  make  an  estimate  of  the  errors  to  be  expected 


102  U.   S.    COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.   14. 

if  the  above  elaborate  plan  of  operations  can  be  carried  out  only  in  part,  as,  for  example,  when 
an  observer  determines  the  longitude  of  a  new  station  by  making  a  single  trip  to  it,  carrying  a 
few  chronometers  only  and  making  all  time  observations  at  both  ends  of  the  trip  himself. 

In  connection  with  any  plan  of  operations  which  involves  long  intervals  between  the 
arrival  at  and  the  departure  from  a  given  station,  it  should  be  kept  in  mind  that  the  computation 
usually  involves  the  assumption  that  the  rates  of  the  traveling  chronometers  are  the  same  on 
the  trip  to  the  station  as  on  the  return  trip,  and  therefore  a  long  stay  at  the  station  is  apt  to 
increase  the  error  of  the  final  result  by  giving  the  chronometers  a  long  time  to  acquire  new  rates. 
Under  extreme  conditions  it  may  sometimes  be  well  to  avoid  this  assumption  and  to  use  a 
separate  traveling  rate  for  each  half  trip  derived  from  observations  just  preceding  or  following 
that  half  trip. 


PART    III. 


THE  DETERMINATION  OF  LATITUDE  BY  MEANS  OF  THE  ZENITH  TELESCOPE. 


INTRODUCTORY. 

A  measurement  of  the  meridional  zenith  distance  of  a  known  star,  or  other  celestial  object, 
furnishes  a  determination  of  the  latitude  of  the  station  of  observation.  In  the  zenith  telescope, 
or  IIorrebow-Talcott,1  method  of  determining  the  latitude,  there  is  substituted  for  the  measure- 
ment of  the  absolute  zenith  distance  of  a  star  the  measurement  of  the  small  difference  of  meridional 
zenith  distances  of  two  stars  culminating  at  about  the  same  time,  and  on  opposite  sides  of  the 
zenith.  The  effect  of  this  substitution  is  the  attainment  of  a  much  higher  degree  of  precision, 
arising  from  the  increased  accuracy  of  a  differential  measurement,  in.  general,  over  the  corre- 
sponding absolute  measurement;  from  the  elimination  of  the  use  of  a  graduated  circle  from  the 
essential  part  of  the  measurement;  and  from  the  fact  that  the  computed  result  is  affected,  not 
by  the  error  in  estimating  the  absolute  value  of  the  astronomic  refraction,  but  simply  by  the 
error  in  estimating  the  very  small  difference  of  refraction  of  two  stars  at  nearly  the  same  altitude. 

Because  of  its  great  accuracy,  combined  with  convenience  and  rapidity,  the  Horrebow- 
Talcott  method  has  become  the  only  standard  method  of  this  Survey.  For  other  methods  of 
determining  the  latitude,  involving  in  most  cases  absolute  measurements  of  zenith  distance  or 
altitude,  the  reader  is  referred  to  treatises  on  astronomy. 

The  method  of  determining  the  latitude  by  observing  the  time  of  transit  of  a  star  across 
the  prime  vertical,  is  one  which  is  capable  of  a  very  high  degree  of  accuracy  and  is  well  adapted 
to  field  use,  as  the  effects  of  instrumental  errors  may  be  readily  eliminated.  To  determine  the 
latitude  of  a  station  by  this  method,  the  times  of  transit  of  various  stars  (of  positive  declination 
less  than  the  latitude)  across  the  plane  of  a  transit  placed  approximately  in  the  prime  vertical 
are  observed.  The  inclination  of  the  transverse  axis  is  determined  accurately  with  a  striding 
level.  The  effects  of  error  of  collimation  and  pivot  inequality  are  eliminated  by  reversal  of  the 
axis.  The  effects  of  azimuth  error  (deviation  of  the  instrument  from  the  prime  vertical)  and 
of  constant  errors  in  the  observed  times  (personal  equation)  are  eliminated  by  observing  some 
stars  to  the  eastward  of  the  zenith  and  others  to  the  westward.  The  declinations  of  the  stars 
observed  must  be  accurately  known,  as  the  declination  errors  enter  directly  into  the  latitude  at 
about  their  full  value,  but  the  right  ascensions  need  be  known  but  approximately. 

This  method  has  been  little  used  by  this  Survey,  perhaps  because  more  time  is  required  to 
prepare  an  extended  observing  list  than  in  the  zenith  telescope  method,  but  it  may  be  found 
useful  in  the  future.  If  the  only  instrument  available  is  a  theodolite  having  a  good  striding 
level,  but  not  equipped  for  observations  by  the  zenith  telescope  method,  observations  in  the 
prime  vertical  will  give  the  best  possible  determination  of  the  latitude.  (For  details  in  regard 
to  this  method,  see  Chauvenet's  Astronomy,  Vol.  II,  pp.  238-271,  and  Doolittle's  Practical 
Astronomy,  pp.  348-377.  For  an  interesting,  early  test  of  the  method  [1827]  by  Bessel,  with 
a  very  small  portable  instrument,  see  Astronomische  Nachrichten,  Vol.  9,  pp.  413-436.) 

GENERAL   INSTRUCTIONS   FOR  LATITUDE  WORK. 

1.  In  order  that  the  records  and  computations  of  the  latitude  work  of  this  Survey  may  be 
uniform  in  character  and  that  there  may  be  approximately  the  same  accuracy  in  the  results, 
some  general  directions  are  given  here  which  should  be  carried  out  by  all  observers  of  this  Survey, 

1  See  p.  245  of  Appendix  14,  Report  for  1880,  for  some  general  remarks  on  Talcott's  method. 

103 


104  U.   S.   COAST  AND  GEODETIC    SURVEY    SPECIAL   PUBLICATION    XO.   14. 

engaged  upon  this  class  of  work,  unless  they  are  directed  otherwise  by  special  instructions  or 
unless  exceptional  circumstances  are  encountered  which  make  changes  necessary  or  desirable. 

2.  The  Horrebow-Talcott  method  should  be  followed,  using  the  zenith  telescope  or  the 
meridian  telescope.     (See  p.  8  for  description  of  the  latter  instrument.     The  zenith  telescope 
is  described  below.) 

3.  A  pair  of  stars  should  be  observed  only  once  at  a  given  station,  unless  some  gross  error 
is  discovered,  in  which  case  the  pair  may  be  reobserved.     Not  more  than  two  stars  should  be 
observed  at  one  setting  of  the  instrument.     A  star  may  be  observed  on  more  than  one  night, 
if  paired  with  a  different  star  on  each  night. 

4.  A  sufficient  number  of  pairs  should  be  observed  at  a  station  to  make  it  reasonably 
certain  that  the  probable  error  of  the  mean  result  is  not  greater  than  ±0".10  (see  directions 
for  procedure  in  making  the  office  computation).     No  additional  expenditure  of  time  or  money 
should  be  made  in  trying  to  reduce  the  probable  error  below  this  limit.     In  no  case,  however, 
should  the  number  of  pairs  observed  at  a  station  be  less  than  10. 

5.  No  determination  of  the  micrometer  value  should  be  made  in  the  field,  as  this  value  is 
computed  at  the  office  from  the  regular  observations  for  latitude. 

6.  The  pairs  observed  should  be  so  selected  that  the  algebraic  sum  of  the  measured  micro- 
meter differences  in  turns  at  a  station  is  less  than  the  total  number  of  pairs.     This  sum  should 
be  made  small,  in  order  that  the  computed  latitude  may  be  nearly  free  from  any  effect  of  error 
in  the  mean  value  of  the  micrometer  screw. 

7.  The  stars  observed  upon  should  be  taken  from  "The  Preliminary  General  Catalogue  of 
6188  Stars  for  the  Epoch  1900"  by  Lewis  Boss,  which  was  published  by  the  Carnegie  Institution 
of  Washington  in  1910. 

8.  Duplicates  of  the  latitude  records,  in  the  form  of  entries  in  the  latitude  computation 
sheets,  should  be  made  and  checked  as  the  work  progresses.     Only  such  portions  of  the  latitude 
computations  should  be  made  in  the  field  as  are  necessary  to  ascertain  the  degree  of  accuracy 
secured. 

9.  The  duplicates  and  computations,  both  complete  and  incomplete,  for  each  station  should 
be  sent  to  the  office  by  registered  mail,  as  soon  as  practicable  after  the  completion  of  the  occu- 
pation of  the  station.     Each  book  of  original  records  should  be  sent  to  the  office  by  registered 
mail  soon  after  the  last  of  the  corresponding  duplicates  and  computations  have  been  forwarded, 
but  not  so  soon  as  to  arrive  in  Washington  by  the  same  mail.     It  is  desirable  to  have  the  records 
and  computations  sent  to  the  office  promptly,  in  order  to  avoid  their  possible  loss. 

10.  Original  descriptions  of  stations  should  be  inserted  in  the  original  record  of  latitude 
observations  and  a  duplicate  description  of  each  station  should  be  written  in  a  volume  kept 
especially  for  the  purpose.     This  volume  should  be  sent  to  the  office  at  the  close  of  a  season's 
work. 

11.  The  form  of  record  of  observations  and  of  field  and  office  computations  of  results 
should  conform  to  those  shown  in  this  publication. 

These  General  Instructions  will  be  referred  to  from  time  to  time  in  the  siicceeding  text. 

DESCRIPTION   OF  THE   ZENITH  TELESCOPE. 

Illustration  No.  13  shows  one  of  the  best  zenith  telescopes  now  in  use  in  this  Survey.  This 
instrument,  Zenith  Telescope  No.  4,  was  originally  made  by  Troughton  &  Simms,  of  London, 
in  1849,  and  was  remodeled  at  the  Coast  and  Geodetic  Survey  Office  in  1891.  It  carries  a 
telescope  with  a  clear  aperture  of  about  76mm  (3*inches),  and  a  focal  length  of  about  116,6cm 
(46  inches).  The  magnifying  power  with  the  eyepiece  ordinarily  used  is  100  diameters.  Two 
latitude  levels  are  used  instead  of  one,  to  secure  increased  accuracy.  Each  of  these  levels 
carries  a  graduation  which  is  numbered  continuously  from  one  end  to  the  other  (instead  of 
each  way  from  the  middle),  the  numbering  of  the  upper  one  running  from  0  to  50  and  of  the 
lower  from  60  to  110.  A  2mm  division  on  the  upper  level  has  a  value  of  about  1".6  and  on  the 
lower  about  1".4.  The  vertical  axis  of  the  instrument  is  in  the  vertical  plane  in  which  the 
telescope  swings.  The  clamp  arm,  perforated  for  the  sake  of  lightness,  gives  the  telescope  a 


No.  13. 


ZENITH  TELESCOPE. 


DETERMINATION   OF   LATITUDE.  105 

marked  degree  of  stability  in  so  far  as  changes  of  inclination  are  concerned.  The  eyepiece 
micrometer,  arranged  to  measure  zenith  distance,  has  a  value  of  about  45"  per  turn,  and  the 
micrometer  head  is  graduated  to  hundredths  of  a  turn. 

The  better  known  type  of  zenith  telescope,  in  which  the  telescope  is  mounted  eccentrically 
on  one  side  of  the  vertical  axis  instead  of  in.  front  of  it,  is  also  in  use  in  the  Survey.  The  meridian 
telescopes  described  on  page  8  are  extensively  used  for  latitude  determinations,  as  well  as 
for  time. 

In  latitude  work  with  the  meridian  circle  at  astronomic  observatories  the  instrument  is 
usually  fitted  with  a  reversing  prism.  By  rotating  this  prism  the  apparent  motion  of  the  star 
is  changed  from  the  direction  right  to  left  to  the  direction  left  to  right  or  vice  versa.  A  pointing 
is  made  on  the  star  before  it  transits,  the  prism  is  reversed,  and  a  second  pointing  is  made  after 
the  transit.  The  observer  may  always  place  the  wire  above  the  center  of  the  star's  image  (or 
below)  but  as  the  image  is  reversed  by  the  prism,  one  of  the  pointings  is  made  on  the  south  side 
of  the  center  of  the  star  and  the  other  pointing  on  the  north  side.  The  mean  of  the  two  point- 
ings will  be  free  from  any  constant  or  systematic  error  in  the  bisection  of  the  star.  It  is  believed 
that  the  systematic  error  of  bisection  does  not  affect  the  results  of  latitude  observations  made 
by  the  Talcott  method,  except  to  a  small  degree  due  to  the  fact  that  an  observer's  systematic 
error  of  bisection  may  be  slightly  different  for  stars  of  different  magnitude.  A  pair  may  be 
composed  of  stars  of  very  different  magnitudes.  The  reversing  prism  need  not  be  used  in  any 
latitude  observations  by  the  Talcott  method  which  are  made  for  the  usual  geodetic  orgeographic 
purposes. 

SUPPORT  FOR  THE   INSTRUMENT. 

The  support  for  the  latitude  instrument  most  frequently  used  in  this  survey  is  a  wooden 
tripod  made  of  lumber  about  6  inches  square  in  cross-section,  well  braced  and  set  firmly  in 
the  ground  to  a  depth  of  from  1  to  3  feet,  depending  on  the  nature  of  the  soil.  Piers  made  of 
brick,  of  cement  blocks,  or  of  concrete  are  also  used.  The  concrete  pier  is  not  as  satisfactory 
as  the  other  types,  if  it  is  used  very  soon  after  it  is  constructed.  When  latitude  and  azimuth 
are  both  observed  at  a  station  the  same  pier  may  be  used  for  mounting  both  the  latitude  instru- 
ment and  the  theodolite.  A  type  of  pier  used  by  some  of  the  parties  of  this  Survey  is  shown 
in  illustration  No.  24  and  is  described  on  page  140. 

OBSERVATORIES  AND   OBSERVING  TENTS. 

At  the  field  stations  only  a  temporary  structure  to  protect  the  instrument  from  wind 
during  the  observations  and  from  rain  during  the  stay  at  the  station  is  needed.  The  observer 
is  seldom  at  a  station  more  than  a  week  after  everything  has  been  made  ready  for  the  observing, 
and  an  observatory  such  as  is  shown  in  illustration  No.  14,  built  of  rough  lumber,  answers  every 
purpose.  It  is  advisable  to  have  2  doors  in  the  observatory  to  insure  the  free  circulation  of 
air.  No  part  of  the  building  should  touch  the  ground  except  at  the  corners.  The  roof  may 
be  made  water-tight  by  boards  or  a  covering  of  felt  or  tar  paper.  A  canvas  sheet  is  sometimes 
carried  with  the  outfit  and  the  roof  is  made  by  stretching  this  sheet  over  the  rafters  and  tying 
it  to  the  sides  of  the  observatory.  The  canvas  may  be  removed  during  the  observations,  thus 
leaving  the  whole  top  of  the  observatory  open  to  the  sky. 

When  a  station  is  located  in  a  town,  although  for  only  a  short  time,  the  observatory  should 
as  a  rule  be  made  neatly,  of  smooth  lumber,  as  shown  in  illustration  No.  15.  Buildings  at 
permanent  latitude  stations  need  not  be  discussed  here,  as  this  publication  deals  only  with 
observations  made  for  geodetic  or  geographic  purposes. 

An  observing  tent  such  as  is  shown  in  illustration  No.  16  or  in  illustration  No.  17  is  more 
frequently  used  on  latitude  work  than  the  wooden  observatory,  and  it  has  the  great  advantage 
that  it  is  easily  transported  and  quickly  set  up.  Except  on  mountain  peaks  or  at  other  places 
where  transportation  is  difficult  the  tent  has  a  floor  similar  to  that  used  with  an  observatory. 

Where  a  floor  or  platform  is  not  used,  the  observer  must  be  extremely  careful  not  to  shift 
his  weight  during  the  interval  between  the  pointing  on  a  star  and  the  reading  of  the  levels. 


106  U.   S.    COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 

and  in  this  case  the  bubble  readings  must  be  made  by  an  attendant  who  must  also  stand  in 
one  place  without  shifting  his  weight  from  the  time  the  observation  is  made  until  the  level 

is  read. 

ADJUSTMENTS. 

When  setting  up  the  instrument  place  two  of  the  foot  screws  in  an  east  and  west  line. 
The  level  correction  may  then  be  kept  small  during  the  progress  of  the  observations  by  using 
one  foot  screw  only. 

The  vertical  axis  may  be  made  approximately  vertical  by  use  of  the  plate  level,  if  there 
is  one  on  the  instrument,  and  the  final  adjustment  made  by  using  the  latitude  level.  The 
position  of  the  horizontal  axis  may  then  be  tested  by  readings  of  the  striding  level.  If  the 
horizontal  axis  is  found  to  be  inclined,  it  must  be  made  horizontal  by  using  the  screws  which 
change  the  angle  between  the  horizontal  and  vertical  axes,  if  the  instrument  is  of  the  old  form. 
With  the  new  form  of  instrument  (illustration  No.  13),  or  with  a  meridian  telescope,  the  two 
axes  will  always  remain  so  nearly  at  right  angles  that  no  means  for  making  this  adjustment  is 
needed.  With  these  instruments  the  vertical  axis  may  be  made  vertical  by  using  both  the 
striding  level  and  the  latitude  level  at  the  same  time. 

The  eyepiece  and  objective  should  be  carefully  focused  as  indicated  on  pages  14  and  15. 
It  is  important  that  the  focus  of  the  objective  should  be  kept  constant  during  the  stay  at  a 
station,  since  the  angular  value  of  one  turn  of  the  eyepiece  micrometer  is  depended  upon  to 
remain  constant  for  the  station.  However,  the  results  of  the  determination  of  the  value  of  a 
turn  of  the  micrometer  vary  in  some  cases  as  much  as  0".13,  corresponding  to  a  range  of  about 
3.3  millimeters  in  the  distance  between  the  objective  and  the  micrometer  lines  (see  p.  129). 
In  connection  with  the  common  habit  of  carefully  keeping  the  draw  tube  clamped  for  the 
purpose  of  holding  the  micrometer  value  constant,  it  is  interesting  to  note  that  while  in  the 
field  in  1905  Assistant  W.  H.  Burger  focused  zenith  telescope  No.  2  five  times  in  rapid  succession 
with  a  range  of  only  0.1  millimeter  in  the  position  of  the  sliding  tube. 

The  movable  micrometer  thread  with  which  all  pointings  are  to  be  made  must  be  truly 
horizontal.  This  adjustment  may  be  made,  at  least  approximately,  in  daylight  after  the 
other  adjustments.  Point,  with  the  movable  thread,  upon  a  distant  well-defined  object,  with 
the  image  of  that  object  near  the  apparent  right-hand  side  of  the  field  of  the  eyepiece,  and  with 
the  telescope  clamped  in  zenith  distance.  Shift  the  image  to  the  apparent  left-hand  side  of 
the  field  by  turning  the  instrument  about  its  vertical  axis.  If  the  bisection  is  not  still  perfect, 
half  the  correction  should  be  made  with  the  micrometer  and  half  with  the  slow-motion  screws 
which  rotate  the  whole  eyepiece  and  reticle  about  the  axis  of  figure  of  the  telescope.  Repeat, 
if  necessary.  The  adjustment  should  be  carefully  tested  at  night  after  setting  the  stops  by 
taking  a  series  of  pointings  upon  a  slow-moving  star  as  it  crosses  the  field  with  the  telescope  in 
the  meridian.  If  the  adjustment  is  perfect,  the  mean  reading  of  the  micrometer  before  the 
star  reaches  the  middle  of  the  field  should  agree  with  the  mean  reading  after  passing  the  middle, 
except  for  the  accidental  errors  of  pointing.  It  is  especially  important  to  make  this  adjustment 
carefully,  for  the  tendency  of  any  inclination  is  to  introduce  a  constant  error  into  the  computed 
values  of  the  latitude. 

The  line  of  collimation  (see  p.  13)  as  defined  by  the  middle  vertical  line  of  the  reticle  must 
be  very  nearly  perpendicular  to  the  horizontal  axis.  If  the  instrument  is  a  meridian  telescope, 
or  of  the  form  shown  in  illustration  No.  13,  this  adjustment  may  be  made  as  for  a  transit  (p.  15) 
by  reversing  the  horizontal  axis  in  the  wyes.  If  the  instrument  is  of  the  form  in  which  the 
telescope  is  to  one  side  of  the  vertical  axis,  the  method  of  making  the  test  must  be  modified 
accordingly.  It  may  be  made  by  using  two  collimating  telescopes  which  are  pointed  upon 
one  another  in  such  positions  that  the  zenith  telescope  may  be  pointed  first  upon  one  and  then 
upon  the  other  with  no  intermediate  motion  except  a  rotation  of  180°  about  the  horizontal 
axis.  It  may  be  made  as  for  an  engineer's  transit,  but  using  two  fore  and  two  back  points, 
the  distance  apart  of  each  pair  of  points  being  made  double  the  distance  between  the  vertical 
axis  and  the  axis  of  collimation  of  the  telescope.  A  single  pair  of  points  at  that  distance  apart 
may  ba  used  and  the  horizintal  circle  trusted  to  determine  when  the  instrument  has  been  turned 


No.  U. 


OBSERVATORY. 


DETERMINATION    OF   LATITUDE.  107 

180°  in  azimuth.  Or  a  single  point  at  an  approximately  known  distance  may  be  used  and  the 
horizontal  circle  trusted  as  before,  and  a  computed  allowance  made  on  the  horizontal  circle 
for  the  parallax  of  the  point  when  the  telescope  is  changed  from  one  of  its  positions  to  the 
other.  Thus,  let  d  =  the  distance  of  the  vertical  axis  from  the  axis  of  collimation  of  the  tele- 
scope, D  =  the  distance  to  the  point,  and  p  =  the  parallax  for  which  correction  is  to  be  made ; 
then,  in  seconds  of  arc: 

2d 
p~Dsml" 

If  one  considers  the  allowable  limit  of  error  in  this  adjustment  (see  p.  134)  it  is  evident  that 
refined  tests  are  not  necessary,  and  that  a  telegraph  pole  or  small  tree,  if  sufficiently  distant  from 
the  instrument,  may  be  assumed  to  be  of  radius  =  d,  and  the  adjustment  made  accordingly. 

The  stops  on  the  horizontal  circle  must  be  set  so  that  when  the  abutting  piece  is  in  contact 
with  either  of  them  the  line  of  collimation  is  in  the  meridian.  For  this  purpose  the  chronometer 
correction  must  be  known  roughly — within  one  second,  say.  Set  the  telescope  for  an  Ephemeris 
star  which  culminates  well  to  the  northward  of  the  zenith,  and  look  up  the  apparent  right 
ascension  for  the  date.  Follow  the  star  with  the  middle  vertical  line  of  the  reticle,  at  first 
with  the  azimuth  motion  free  and  afterwards  using  the  tangent  screw  on  the  horizontal  circle, 
until  the  chronometer,  corrected  for  its  error,  indicates  that  the  star  is  on  the  meridian.  Then 
clamp  a  stop  in  place  against  the  abutting  piece.  Repeat  for  the  other  stop,  using  a  star  which 
culminates  far  to  the  southward  of  the  zenith.  It  is  well,  if  time  permits,  to  test  the  setting 
of  each  stop  by  an  observation  of  another  star  before  commencing  latitude  observations. 

The  correction  to  the  chronometer  may  be  obtained  by  observations  on  the  sun  or  stars 
with  a  sextant  or  a  vertical  circle  (see  pp.  52-56),  by  observing  the  time  of  transit  of  stars  with  a 
theodolite,  or  by  using  the  zenith  telescope  as  a  transit  instrument.  With  the  zenith  telescope 
in  good  adjustment  and  approximately  in  the  meridian  and  the  sidereal  time  known  within 
several  minutes,  the  chronometer  time  of  transit  of  a  star  near  the  zenith  is  noted.  This  obser- 
vation gives  a  close  approximation  to  the  chronometer  error.  Then  a  north  star  of  high  decli- 
nation is  used  and  the  telescope  is  put  more  nearly  in  the  meridian  by  the  method  explained 
above.  Next  the  chronometer  time  of  transit  of  a  second  zenith  star  is  observed,  which  will 
usually  give  the  chronometer  correction  within  a  second.  With  this  value  of  the  chronometer 
correction  the  telescope  may  be  put  closely  enough  in  the  meridian  for  observing. 

The  finder  circle  must  be  adjusted  to  read  zenith  distances  (see  p.  16). 

THE  OBSERVING  LIST. 

The  Boss  catalogue1  of  6188  stars  is  now  available,  and  is  at  present  the  best  list  from 
which  to  select  pairs  of  stars.  (See  paragraph  7  of  General  Instructions,  p.  104.)  The  latitude 
of  the  station  should  be  obtained  to  the  nearest  minute  from  a  map,  a  triangulation  station,  or 
from  preliminary  observations  on  the  sun  or  stars.  In  the  Boss  catalogue  the  declinations  of 
the  stars  are  given  and  the  observing  list  may  be  made  out  like  the  form  shown  below.  Any 
other  arrangement  of  the  data  may  be  used.  To  find  all  available  pairs  in  a  given  list  one  may, 
for  each  star  in  succession  within  the  zone  of  observation,  45°  each  way  from  the  zenith,  sub- 
tract the  declination  from  twice  the  latitude  and  then  compare  this  difference  with  the  decli- 
nation of  each  star  in  the  list  within  the  following  20m  of  right  ascension.  Any  star  whose 
declination2  is  within  20'  of  the  above  difference  will  combine  with  the  star  under  considera- 
tion to  make  a  pair,  provided  the  other  conditions  stated  below  are  fulfilled.  By  proceeding 
thus  every  available  pair  will  be  found.3 

1  Preliminary  general  catalogue  of  6188  stars  for  the  epoch  1900,  Lewis  Boss,  Carnegie  Institution  of  Washington,  1910. 

2  Or  180°— t  for  subpolars. 

3  At  stations  in  Alaska  there  are  but  few  stars  in  the  zone  extending  45°  northward  from  the  zenith  as  compared  with  the  corresponding  zone 
to  the  southward,  and  the  above  process  may  be  improved  by  taking  in  succession  only  stars  to  the  north  of  the  zenith  and  comparing  each  with 
stars  in  both  the  preceding  and  the  following  10™.    To  make  the  search  with  a  subpolar  star  subtract  180°— 3  from  twice  the  latitude  and  pair  with 
any  star  whjse  declination  is  within  2ff  of  this  difference,  provided  its  right  ascension  differs  from  that  of  the  subpolar  anywhere  from  llh  40"  to 
12">  20». 


108 


U.    S.   COAST   AND  GEODETIC    SURVEY    SPECIAL   PUBLICATION    NO.   14. 


Observing  list  (Form  1). 

[St.  Anne,  111.,  June  23,  1908.    Zenith  telescope  No.  4.    ^=41°  Ol'.S.    Search  faetor=2  0=82°  03'.] 


Star  No. 
Boss 
catalogue 

Mag. 

Right  ascen- 
sion 

Declina- 
tion 
S 

Differ- 
ence 
between 

3'a 

£»= 

sum  of 
declina- 
tions 

Zi-'ij, 

N-S- 
0*  (Xi— 

if) 

Star 
north  or 
south 

Setting 
=  j  differ- 
ence of 

3's 

Tunis 

h      m       s 

o         / 

0             / 

0             / 

f 

0             / 

4327 

4.5 

16     55     22 

82     11 

N 

12 

4379 

4.9 

17     11     53 

-0     21 

82    32 

81     50 

-13 

-17 

S 

41     16 

28 

4441 

5.9 

17     28     13 

28    28 

S 

10 

4494 

5.8 

17     42    04 

53    50 

25    22 

82    18 

+15 

+20 

N 

12     41 

30 

4623 

5.1 

18     13    22 

64    22 

N 

24 

4651 

5.4 

18    18    45 

17     47 

46    35 

82    09 

+  6 

+  8 

S 

23     18 

16 

4669 

5.9 

18    22    26 

29    47 

S 

20 

4711 

5.5 

18    31    52 

52     17 

22     30 

82    04 

+  1 

+  1 

N 

11     15 

20 

*  a= number  of  turns  of  the  micrometer  screw  in  one  minute  of  arc=1.34.    The  value  of  one  turn  of  the  micrometer  screw=44".650. 

The  approximate  mean  right  ascensions  and  declinations  for  the  observing  list  are  obtained 
for  the  time  of  the  observations  by  multiplying  the  annual  variation  by  the  number  of  years 
elapsed  since  the  epoch  of  the  catalogue  and  combining  the  products  algebraically  with  the 
right  ascension  and  declination  given  in  the  catalogue  used. 

In  the  above  form  there  is  no  column  for  zenith  distances.  The  setting  for  a  pair  is  one- 
half  the  difference  between  the  declinations  of  the  two  stars  of  a  pair.  To  get  the  values  in  the 
column  N  — S  subtract  double  the  latitude  (for  station  St.  Anne,  82°  03')  from  the  sum  of  the 
declinations  of  the  two  stars  and  multiply  the  result  in  minutes  of  arc  by  the  number  of  turns 
of  the  micrometer  screw  in  a  minute  of  arc.  N  — S  is  positive  if  the.  north  star  has  the  greater 
zenith  distance  and  is  negative  if  the  south  star  has  the  greater  zenith  distance.  The  center 
of  the  comb  in  the  micrometer  eyepiece  is  called  20,  and  increasing  readings  on  the  graduated 
head  go  with  increasing  zenith  distances.  Then  the  setting  of  the  micrometer  wire  for  any 

north  star  is  20  H ~ —  and  for  any  south  star  20  —  ^—2 —     These  settings  are  given  in  the  last 

column  of  the  above  table. 

When  one  star  of  the  pair  is  a  subpolar,  the  finder  circle  setting  is  90°  —  \Zd.  N  —  S  in  this 
case  is  a  (180°  — difference  of  d's  —  2</>)  and  is  positive  or  negative  according  as  the  north  star 
has  the  greater  or  lesser  zenith  distance.  The  setting  of  the  micrometer  wire  will  be  given  by 
the  same  general  expression  as  above. 

For  the  purposes  of  the  observing  list  it  is  sufficiently  accurate  to  know  the  mean  right 
ascensions  to  within  one  second  and  the  declinations  and  derived  quantities  to  the  nearest 
minute  of  arc.  The  approximate  reading  of  the  turns  is  given  to  facilitate  identification  of 
the  stars  and  to  enable  the  observer  to  put  the  micrometer  line  approximately  in  position  before 
the  star  enters  the  field  of  view.  The  middle  reading  of  the  micrometer  comb  is  called  20  to 
avoid  negative  readings. 

If  the  Ten  Year  Catalogues  for  1880  and  1890  and  the  Nine  Year  Catalogue  for  1900,  by  the 
Royal  Observatory  at  Greenwich,  are  used,  then  the  form  of  the  observing  list  could  be  made 
to  advantage  in  a  manner  somewhat  different  from  that  shown  above,  for  in  those  publications 
the  north  polar  distances  are  given  instead  of  the  declinations.  The  list  may  be  similar  to  that 
shown  below,  where  the  settings,  etc.,  are  derived  from  the  north  polar  distances  of  the  stars. 
In  the  first  column  of  the  example  are  given  the  Boss  catalogue  numbers,  though  the  stars  are 
also  in  the  lists  of  the  Greenwich  catalogues  mentioned  above.  They  are  the  same  stars  as 
those  in  the  first  form  of  star  list. 


No.  16. 


OBSERVING  TENT. 


No.  17. 


OBSERVING  TENT. 


DETERMINATION   OF   LATITUDE. 

Observing  List  (Form  2). 

[St.  Anne,  HI.,  June  25,  1908.    Zenith  Telescope  No.  4.    j>—  41°  01' .3.    Search  factor- 180*- 2  ^-97°  57*.) 


109 


Sum  of 

Star  No., 
Boss  cat- 
alogue 

Mag. 

a 

North  polar 
distances  and 
difference 

N.  P.  D.'s; 
and  search 
factor  minus 

sum  <if 

N-S* 

Star 
north 
or 
south 

Setting 
=J  dif.  of 
N.  P.  D.'s 

Turns 

N.  P.  D.'s 

o       / 

0          / 

0        / 

h    ffi     s 

4327 

4.5 

16    55    22 

7     49 

N 

12 

4379 

4.9 

17     11    53 

90    21 

98     10 

S 

41     16 

28 

82    32 

-13 

-17 

4441 

5.9 

17    28    13 

61    32 

S 

10 

4494 

5.8 

17     42    04 

36    10 

97    42 

N 

12     41 

30 

25    22 

+15 

-1-20 

4623 

5.1 

18    13    22 

25    38 

N 

24 

4651 

5.4 

18    18    45 

72     13 

97    51 

S 

.    23     18 

16 

46    35 

+  6 

+  8 

4669 

5.9 

18    22    26 

60    13 

S 

20 

4711 

5.5 

18    31    52 

37    43 

97    56 

N 

11     15 

20 

22    30 

+  1 

+  1 

*  N— S— a   (search  factor— sum  of  N.  P.  D.'s),  where  a— number  of  turns  of  the  micrometer  screw  in  one  minute  of 
arc— 1.34.    The  value  of  one  turn  of  the  micrometer  screw— 44". 650. 

When  a  subpolar  star  is  used  slight  changes  will  be  necessary,  similar  to  those  described 
for  the  case  where  the  observing  list  is  prepared  in  terms  of  the  decimations. 

Among  the  requisites  for  a  pair  of  stars  for  an  observing  list,  are,  that  their  right  ascensions 
shall  not  differ  by  more  than  20m,  or  12b±20m  when  a  subpolar  is  used,  to  avoid  too  great  errors 
arising  from  instability  in  the  relative  positions  of  different  parts  of  the  instrument;  nor  by 
less  than  about  lm,  that  interval  being  required  to  take  the  readings  upon  the  first  star  and 
prepare  for  the  second  star  of  a  pair;  that  their  difference  of  zenith  distances  shall  not  exceed 
the  half  length  of  the  micrometer  comb,  20'  for  many  instruments;  that  each  star  shall  be 
bright  enough  to  be  seen  distinctly,  not  fainter  than  the  seventh  magnitude  for  the  larger  instru- 
ments; and  that  no  zenith  distance  shall  exceed  45°,  to  guard  against  too  great  an  uncertainty 
in  the  refraction.  The  third  of  the  above  conditions  may  be  used  more  converiently  in  this 
form;  the  sum  of  the  two  declinations  must  not  differ  from  twice  the  latitude  by  more  than  20'. 
The  total  range  of  the  list  in  right  ascension  is  governed  by  the  hours  of  darkness  on  the  pro- 
posed dates  of  observation. 

In  the  list  of  pairs  resulting  directly  from  the  search  there  will  be  many  pairs  which  overlap 
in  time.  A  feasible  observing  list  may  be  formed  by  omitting  such  pairs  that  among  the 
remainder  the  shortest  interval  between  the  last  star  of  one  pair  and  the  first  star  of  the  next 
is  not  less  than  2m.  In  that  interval  a  rapid  observer  can  finish  the  readings  upon  one  pair  and 
set  for  the  next,  under  favorable  circumstances.  The  omitted  pairs  may  be  included  in  a  list 
prepared  for  the  second  or  third  night  of  observation.  It  will  frequently  be  found  that  the 
same  star  occurs  in  two  or  more  different  pairs.  Such  pairs  may  be  treated  like  those  which 
overlap  in  time.1 

DIRECTIONS  FOR  OBSERVING. 

All  adjustments  having  previously  been  made,  set  for  the  first  star  and  await  it  with  the 
bubble  of  the  latitude  level  nearly  in  the  middle  of  the  tube,  and  with  the  micrometer  line  at 
that  part  of  tho  comb  at  which  the  star  is  expected,  as  shown  by  the  observing  list.  Watch 
the  chronometer  so  as  to  know  when  to  expect  the  star.  When  the  star  enters  the  field,  place 
the  micrometer  line  approximately  upon  it.  As  soon  as  the  star  comes  within  the  safe  observing 
limits  of  the  field  bisect  it  carefully.  As  the  star  moves  along  watch  the  bisection  and  correct 

1  Past  records  furnish  abundant  evidence  that  observations  made  by  pointing  twice  upon  a  close  zenith  star,  once  In  each  position  of  the 
Instrument,  give  results  of  a  low  order  of  accuracy,  probably  because  of  the  hurry  with  which  the  observations  must  bo  made,  and  of  the  fact  that 
one  or  both  of  the  observations  must  be  made  out  of  the  meridian.  It  is  therefore  not  advisable  to  make  such  observations. 


110  U.   S.    COAST  AND  GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 

it  if  any  error  is  detected.  Because  of  momentary  changes  in  the  refraction,  the  star  will 
usually  be  seen  to  move  along  the  line  with  an  irregular  motion,  now  partly  above  it  and  now 
partly  below.  The  mean  position  of  the  star  is  to  be  covered  by  the  line.1  It  is  possible,  but 
not  advisable,  to  make  several  bisections  of  the  star  while  it  is  passing  across  the  field.  As 
soon  as  the  star  reaches  the  middle  vertical  line  of  the  diaphragm  read  off  promptly  from  the 
comb  the  whole  turns  of  the  micrometer,  read  the  level,  and  then  the  fraction  of  a  micrometer 
turn,  in  divisions,  from  the  micrometer  head.  Set  promptly  for  the  next  star,  even  though  it 
is  not  expected  soon.  In  setting  for  the  second  star  of  a  pair  all  that  is  necessary  is  to  reverse 
the  instrument  in  azimuth  and  set  the  micrometer  line  to  a  new  position.  The  abutting  piece 
must  be  brought  gently  against  the  stop  and  the  circle  securely  clamped  in  that  position. 

Especial  care  should  be  taken  in  handling  the  micrometer  screw,  as  any  longitudinal  force 
applied  to  it  produces  a  flexure  of  the  telescope  which  tends  to  enter  the  result  directly  as  an 
error.  The  last  motion  of  the  micrometer  head  in  making  a  bisection  should  always  be  in  one 
direction  (preferably  that  in  which  the  screw  acts  positively  against  its  opposing  spring),  to  insure 
that  any  lost  motion  is  always  taken  up  in  one  direction.  The  bubble  should  be  read  promptly, 
so  as  to  give  it  as  little  time  as  possible  to  change  its  position  after  the  bisection.  The  desired 
reading  is  that  at  which  it  stood  at  the  instant  of  bisection.  Avoid  carefully  any  heating  of 
the  level  by  putting  the  reading  lamp,  warm  breath,  or  face  any  nearer  to  it  than  necessary. 
During  the  observation  of  a  pair  the  tangent  screw  of  the  setting  circle  must  not  be  touched, 
for  the  angle  between  the  telescope  and  the  level  must  be  kept  constant.  If  it  is  necessary 
to  relevel,  to  keep  the  bubble  witliin  reading  limits,  use  the  tangent  screw  which  changes  the 
inch' nation  of  the  telescope.  Even  tliis  may  introduce  an  error,  due  to  a  change  in  the  flexure 
of  the  telescope,  and  should  be  avoided  if  possible.  It  is  desirable  to  relevel  the  instrument 
from  time  to  time  between  pairs,  so  as  to  keep  the  level  correction  small,  less  than  one  division 
of  the  level  if  possible. 

Occasionally  the  approximate  time  should  be  noted  at  which  the  star  being  observed 
crosses  the  middle  vertical  line  of  the  diaphragm,  so  as  to  make  sure  that  the  adjustment  of  the 
stops  in  azimuth  remains  satisfactory.  It  is  desirable  (though  not  necessary)  to  have  a 
recorder.  He,  should  be  a  man  above  the  average  in  intelligence,  and  should  be  able  to  pre- 
pare an  observing  list  after  a  little  practice  and  to  assist  in  computing  the  results.  It  is  not 
economical  to  take  a  man  from  place  to  place  unless  he  can  assist  in  the  computations.  The 
recorder  may  count  seconds  aloud  from  the  face  of  the  chronometer  in  such  a  way  as  to  indicate 
when  the  star  is  to  culminate.  Such  counting  aloud  serves  a  double  purpose.  It  is  a  warning  to 
the  observer  to  be  ready  and  it  indicates  where  to  look  for  the  star  if  it  is  faint  and  difficult  to 
find.  It  also  gives  for  each  star  a  rough  check  upon  the  position  of  the  azimuth  stops.  It  is 
only  a  rough  check,  because  the  observing  list  gives  mean  right  ascensions  instead  of  apparent 
right  ascensions  for  the  date,  but  it  is  sufficiently  accurate  (see  p.  1 19).  The  observer,  or  recorder, 
can  easily  make  allowance  for  the  fact  that  all  stars  (except  circumpolars)  will  appear  to  be  too 
early  or  too  late,  according  to  the  observing  fist,  by  about  the  same  interval,  0s  to  5s,  the  differ- 
ence between  the  mean  and  apparent  right  ascension.  If  a  star  can  not  be  observed  upon  the 
middle  fine,  on  account  of  temporary  interference  by  clouds  or  tardiness  in  preparing  for  the 
observation,  it  may  be  observed  anywhere  witliin  the  safe  limits  of  the  field  (often  indicated 
by  vertical  fines  on  the  diaphragm)  and  the  chronometer  tune  of  observation  recorded.  In 
practice  a  star  is  seldom  observed  off  the  meridian. 

It  is  desirable  to  make  all  settings  with  such  accuracy  that  the  mean  of  the  two  micrometer 
readings  on  a  pair  shall  not  differ  from  20  turns  by  more  than  1  turn.  It  is  not  infrequently 
true  that  the  value  of  a  micrometer  screw  increases  slightly  but  steadily  from  one  end  to  the 
other.  In  such  cases  the  correction  to  each  observed  value  of  the  latitude,  due  to  this  irregu- 
larity of  the  screw,  will  be  insensible  if  the  settings  are  made  with  the  indicated  accuracy,  but 
not  otherwise. 

1  This  wording  must  be  modified  to  correspond  if,  in  accordance  with  the  considerations  stated  on  p.  141,  two  close  parallel  lines  are  used 
Instead  of  a  single  line. 


Form  255. 


DETERMINATION   OF    LATITUDE. 

EXAMPLE   OF  RECORD   AND   COMPUTATIONS. 

Zenith  telescope  record  for  latitude. 

[Station,  St.  Anne.    Date,  June  25, 190S.    Chronometer,  2637.    Observer,  W.  Bowie.] 


Ill 


No.  of 
pair 

Star 
number 
Boss 
Cat. 

N.or 

S. 

Micrometer 

Level 

Chronome- 
ter time  of 
culmina- 
tion 

Chronome- 
ter time  of 
observa- 
tion 

Meridian 
distance 

Remarks 

Turns 

Div's. 

North 

South 

t 

d 

4327 

N 

11 

69.0 

6.0 

39.1 

(*) 

16    55    24 

(*) 

+30f 

9 

67.8 

99.5 

+46 

4379 

S 

27 

34.4 

40.2 

7.2 

17    11    47 

Struck  instrument 

100.5 

68.7 

4441 

S 

9 

61.0 

40.3 

7.2 

17    28    07 

10 

101.2 

69.4 

+24 

4494 

N 

31 

47.0 

7.1 

40.4 

17    41     58 

69.4 

101.3 

4623 

N 

24 

88.2 

9.2 

42.6 

18    13     18 

11 

71.6 

103.8 

+  16 

4651 

S 

16 

66.0 

42.2 

8.7 

18    18    39 

103.2 

71.0 

4669 

S 

19 

62.5 

44.2 

10.9 

18    22    20 

12 

106.0 

73.8 

+  15 

4711 

N 

20 

55.4 

11.2 

44.7 

18    31     45 

Mean  of  double  star 

74.4 

106.5 

Form  32a. 


*  These  columns  are  only  used  when  a  star  is  observed  off  the  meridian. 

t  This  is  the  continuous  sum,  up  to  this  pair,  of  the  south  minus  the  north  micrometer  turns. 

Reduction,  mean  to  apparent  declination,  with  Cape  tables. 

[Station,  St.  Anne  triangulation  latitude  station.] 


Order 
1 
2 

Date 
Star  No. 

June  25,  19C 
4327 

8 
4379 

4441 

4494 

4623 

4651 

4669 

3 

a. 

16     55.  4 

17     11.9 

17    28.  2 

17     42.1 

18     13.  4 

18    18.  7 

18     22.4 

7 
11 
4 

G^r0 

H+aa 
So 

18    33.  8 
4     40.0 
82     11 

18    50.  3 
4    56.5 
-0    20 

19     06.  6 
5     12.8 
28    28 

19    20.  5 
5    26.7 
53    50 

19     51.  8 
5    58.0 
64    21 

19     57.  1 
6     03.3 
17     46 

20    00.8 
6    07.0 
29    46 

8 
9 

P' 
P'x 

+2.95 
-2.49 

+4.36 
-3.68 

+5.74 
-4.84 

+6.90 

-5.82 

+9.40 

-7.92 

+9.81 
-8.27 

+10.  08 
-  8.50 

12 
13 

<?'„ 
8Q' 

+6.26 
0.00 

0.00 
-0.03 

+1.78 
+0.03 

+2.14 
+0.02 

+0.14 
0.00 

-0.08 
0.00 

-  0.27 
0.00 

14 
15 

Q' 
Q'y 

+6.26 
+0.66 

-0.03 
0.00 

+1.81 
+0.19 

+2.16 
+0.23 

+0.14 
+0.01 

-0.08 
-0.01 

-  0.27 
-  0.03 

5 
10 
16 
17 

18 

*o(") 

P'+P'x 

Q'+Q'y 
v-'     V 

23.30 
+0.46 
+6.92 
+0.08 
-.001     .00 

-30.  91 
+  0.68 
-  0.03 
+  0.58 
-.059-.  03 

24.93 
+0.90 
+2.00 
+0.52 
+.  024+.  01 

23.54 
+1.08 
+2.39 
+0.  35 
-.035-.  02 

57.64 
+1.48 
+0.15 
+0.25 
+  .029+.01 

46.61 
+1.54 
-0.09 
+0.56 
+.007     .00 

31.42 
+  1.58 
-  0.30 
+  0.51 
-.033  -.02 

19 

»(") 

30.76 

29.71 

28.36 

27.34 

59.53 

48.62 

33.19 

log  ^=l 
ffo 


log  g=0.  49813 
log  (70=1.30216 

(l+z)=9.  19597 
l+z=+0.  157 


log  h=l.  31041 

log  /!„=!.  26717 

log  ^=log  (1+2,)=0.  04324 
l+y=+1.105 

h          m 
G=  1     38.4 
H=U    44.6 

j=  +0.585 
T=    0.  483 

Make  computation  by  horizontal  lines  in  the  order  indicated.  For  explanation  of  (?„'  and  S  Q',  see  pages  (2)  and  (5) 
of  Cape  tables.  Opposite  S0  in  the  sixth  line  place  the  degrees  and  minutes,  and  opposite  <50  (")  the  seconds  of  the 
mean  declination.  The  quantities  x,  y,  i,  and  T  may  be  assumed  constant  fora  night,  and  should  be  taken  for  an  epoch 
midway  between  the  first  and  last  stars.  The  quantities  G  and  H  may  be  assumed  constant  for  periods  not  exceeding 
four  hours  each,  and  should  be  taken  for  the  midway  epoch  of  each  such  period.  Use  aa,  G,  H.G-\-n0.  and  H-\-a0,  to  tenths 
of  minutes  of  time;  x,  y,  and  T  to  three  significant  figures;  and  all  other  quantities  to  two  decimal  places. 


112 

Form  33. 


U.   S.   COAST   AND   GEODETIC    SURVEY    SPECIAL   PUBLICATION    NO.   14. 


Latitude 

[Station,  St.  Anne.    State,  Illinois. 


Date 

Catalogue                           Micrometer                                            Level 

Meridian 

distance 

Declination 

Star  No. 

Xor 

s 

Reading 

Diff.  Z.D. 

n 

s 

Diff. 

190S. 

t       d 

t            d 

d 

d 

d 

S 

o       /           // 

June  25 

4327 

N 

11     69.0 

06.0 

39.  1 

82     .1  1     30.  76 

+  15    65.4 

67.8 

99.5 

4379 

S 

27    34.4 

40.2 

07.2 

+2.1 

-0     20    29.  71 

100.5 

68.7 

4441 

S 

9     61.0 

40.3 

07.2 

28    28    28.  36 

-21    86.  0 

101.2 

69.4 

4494 

N 

31     47.0 

07.1 

40.4 

-0.05 

53     50    27.  34 

69.4 

101.3 

4623 

N 

24    88.  2 

09.2 

42.6 

64     21     59.53 

-  8     22.2 

71.6 

103.8 

4651 

S 

16    66.0 

42.2 

08.7 

-1.05 

17     46     48.62 

103.2 

71.0 

4669 

S 

19     62.  5 

44.2 

10.9 

29     46     33.  19 

92.9 

106.0 

73.8 

4711 

N 

20     55.  4 

11.2 

44.7 

-0.  95 

52     16     49.44 

74.4 

106.5 

DETERMINATION   OF   LATITUDE. 


113 


computation, 

Observer,  W.  Bowie.    Instrument,  zenith  telescope  No.  4.] 


Sum  and  half  sum 

Corrections 

Latitude 

Remarks 

Micrometer 

Level 

Refraction 

Meridian 

o        /           // 

/           // 

// 

// 

// 

0           /                // 

81    51    01.  05 
40    55    30.  52 

+5    49.48 

+0.78 

+0.18 

41     01     20.  96 

Struck  instrument 

82    18    55.  70 
41     09     27.  85 

-8    08.02 

-0.02 

-0.14 

41    01     19.  67 

82    08    48.  15 
41     04     24.  08 

-3    03.  56 

-0.39 

-0.06 

41    01    20.  07 

82    03     22.  63 
41     01     41.32 

-0    20.  74 

-0.35 

-0.01 

41    01     20.  22 

Value  of  one  division  of  latitude  level :  Upper  —  1.  600 

Lower  -1.364 
Mean    -1.482 


8136°— 13 8 


114 


U.   S.   COAST   AND  GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 

Summary  of  latitude  computation. 

[St.  Anne,  111.,  June  25,  1908.] 


Star  No. 

Mic.  diff. 

* 

u.                      —:"• 

Corrected 

—  2 

Boss  catalogue 

M 

41°  01' 

6$                      J0 

* 

jiji 

3667 

3729 

+  8.3 

20.26 

-0.03 

0.00 

-0.11 

20.15 

+0.09 

0.01 

*(2265) 

3803 

+  0.1 

19.77 

+0.46 

0.21 

0.00 

19.77 

+0.47 

0.22 

3842 

3856 

-14.3 

20.02 

+0.21 

0.04 

+0.20 

20.22 

+0.02 

0.00 

3949 

3979 

+14.1 

20.40 

-0.17 

0.03 

-0.19 

20.21 

+0.03 

0.00 

4063 

4072 

-  9.1 

20.24 

-0.01 

0.00 

+0.12 

20.36 

-0.12 

0.01 

4081 

4090 

+13.8 

20.54 

-0.31 

0.10 

-0.19 

20.35 

-0.  11 

0.01 

4112 

4129 

+16.3 

20.15 

+0.08 

0.01 

-0.22 

19.93 

+0.31 

0.  10 

4161 

4201 

+  2.3 

19.80 

+0.43 

0.18 

-0.03 

19.77 

+0.47 

0.22 

4327 

4379 

+15.7 

20.96 

-0.73 

0.53 

-0.22 

20.74 

-0.50 

0.25 

4441 

4494 

-21.9 

19.67 

+0.56 

0.31 

+0.30 

19.97 

+0.27 

0.07 

4623 

4651 

-  8.2 

20.07 

+0.  16 

0.03 

+0.11 

20.18 

+0.06 

0.00 

4669 

4711 

-  0.9 

20.22 

+0.01 

0.00 

+0.01 

20.23 

+0.01 

0.00 

4745 

4758 

+  2.8 

20.77 

-0.54 

0.29 

-0.04 

20.73 

-0.49 

0.24 

*(3019) 

4799 

-16.4 

20.53 

-0.30 

0.09 

+0.22 

20.75 

-0.51 

0.26 

4824 

4892 

-16.1 

20.06 

+0.17 

0.03 

+0.22 

20.28 

-0.04 

0.00 

73  4 

2  08 

1  85 

1  73 

1  39 

86  9 

2  09 

1  77 

Algebraic  sum 

-13.5 

-0.01 

Mean 

-  0.9 

20.23 

20  24 

*  2285  and  3019  are  ten-year  1880  numbers.    The  mean  declinations  for  these  stars  were  obtained  from  several  sources. 


O.  455X1.85       ,  n/,  9f- 

^  -— 


The  value  of  one-half  turn  of  the  micrometer  as  used  in  the  field  =  22".325. 
Mean  <f>,  8  pairs  with  plus  micrometer  difference  =  41°  01'  20".33. 
Mean  <f>,  7  pairs  with  minus  micrometer  difference  =  41°  01'  20". 12. 

The  mean  of  7  pairs  with  minus  micrometer  differences  minus  the  mean  of  8  pairs  with  plus 
micrometer  differences  =  —  0".21. 


Normal  equations 

15c+13.  5^-0.01=0 

13.  5c+2346.  59^+31.  872=0 

r,=--0".0187 

c=+0".  0130 


Observation  equations 
c-  8.  3)-! -0.03=0 
c-  0.1^+0.46=0 
c+14.  S^+0. 21=0 
c-14. 1^-0. 17=0 
c+  9.1^-0.01=0 
c-13.  8^-0. 31=0 
c-16.  Srj+0. 08=0 
c-  2.3^+0.43=0 
c-15.  7^-0.  73=0 
c+21.  9^+0. 56=0 
c+  8.  27-j+0. 16=0 
c+  0.  9^+0. 01=0 
c-  2. 8r, -0.54=0 
c+16. 4^-0. 30=0 
c+16.  lr,+0. 17=0 

Latitude  of  St.  Anne  latitude  station 
Reduction  to  sea  level,  elevation  of  station,  206  meters 
Reduction  to  mean  position  of  pole * 

Latitude  of  St.  Anne  latitude  station,  reduced  to  sea  level  and 
the  mean  position  of  the  pole 

For  an  explanation  of  the  above  adjustment  see  page  130. 


^W'«?=±o^ 

Corrected   value  of  one-half  turn  of  micrometer  screw 
=22".  SllSiO".  0046 


eB=±^/0455Xl139=±0.22 


=  41°  01'  20".24±0".06 
-0.03 
+  0.07 

=  41°  01'  20".28±0".06 


1  See  Astroaomische  Nachrichten  No.  4414. 


DETERMINATION   OF   LATITUDE.  115 

GENERAL   NOTES   ON   COMPUTATIONS    OF   LATITUDE    IN   THE  UNITED  STATES  COAST  AND 

GEODETIC  SURVEY. 

The  result  from  each  pair  of  stars  is  given  equal  weight.  This  is  done  upon  the  supposition 
that  the  theoretical  weights  are  so  nearly  equal  that,  if  they  were  used,  the  final  value  for  the 
latitude  of  a  station  would  seldom  be  changed  by  more  than  0".01. 

A  first  rejection  limit  of  3 ".00  from  the  mean  value  of  the  latitude  is  used.  After  the 
3".00  rejection  limit  has  been  applied  the  probable  error  of  a  result  from  a  single  pair,  ep,  is 
computed  from  all  the  remaining  values,  and  then  5ep  is  used  as  an  absolute  rejection  limit, 
and  3.5ep  is  used  as  a  doubtful  limit  beyond  which  rejection  is  to  be  made  if  strong  evidence  in 
favor  of  rejection  is  found  other  than  the  residual  itself.  Such  evidence  may  consist  of  positive 
notes  indicating  bad  conditions  during  the  observation  of  the  particular  pair  concerned,  con- 
tradictions in  the  record  indicating  a  probable  misreading,  or  a  mean  declination  of  a  star  with 
a  probable  error  so  large  that  it  might  account  for  the  large  residual. 

A  new  value  of  one-half  turn  of  the  micrometer  is  to  be  derived  from  the  latitude  observa- 
tions only  in  those  cases  in  which  the  mean  latitude  from  pairs  with  plus  micrometer  differ- 
ences differs  by  more  than  0".20  from  the  mean  latitude  from  pairs  with  minus  micrometer 
differences.  It  is  believed  that,  when  the  agreement  is  within  0".20,  a  new  value  of  one-half 
turn,  if  derived  from  the  observations,  would  differ  from  the  old  by  less  than  0".01  and  the 
final  latitude  would  ordinarily  be  changed  by  less  than  0".01.  It  is  also  believed  that  the  derived 
correction  to  the  old  value  would,  in  these  cases,  be  but  little,  if  any,  larger  than  its  own  probable 
error. 

The  formulae  used  in  computing  the  probable  errors,  if  a  correction  to  the  micrometer  value 
is  derived  from  the  latitude  observations,  are: 


1(0. 

,=Y- 


(p-2) 


«# 


V(0.455)2J</>2 
(p-2)(p-^$ 

V  ' 


(0.455)2"  J^. 
er  =  probable  error  of  r.=        -       —-  —       — 


(p_2)jjfl> 

The  correction  for  elevation  to  reduce  the  mean  latitude  to  sea  level  is  always  applied. 
(See  p.  130.) 

The  reduction  to  a  triangulation  station  or  to  other  points  is  also  applied  on  the  latitude 
computation  and  the  relation  of  the  latitude  station  to  such  point  or  points  is  there  indicated. 
Unless  the  latitude  station  is  within  a  few  meters  of  the  triangulation  station  and  due  east  or 
west  of  it,  the  latitude  computation  should  show  the  latitude  of  both  the  latitude  station  and 
the  triangulation  station. 

EXPLANATION  OF  COMPUTATION. 

Let  £  and  £'  equal  the  true  meridional  zenith  distances  of  the  southern  and  northern  stars, 
and  8  and  8'  the  apparent  declinations  of  the  same,  respectively;  then  the  expression  for  the 
latitude  is 


Now,  if  z,  z'  denote  the  observed  zenith  distances  of  the  south  and  the  north  stars;  n,  s  the 
north  and  the  south  readings  of  the  level  for  the  south  star,  and  n'  ,  s'  the  same  for  the  north 
star;  d  the  value  of  one  division  of  level;  r  and  r'  the  refraction  corrections  and  m  and  m'  the 


116  U.    S.    COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 

reductions  of  the  measured  zenith  distances  to  the  meridian  for  the  south  and  the  north  stars, 
respectively,  then 

<p= 

and  if  JWand  M'  be  the  micrometer  readings  of  the  south  and  the  north  stars,  increased  microm- 
eter readings  corresponding  to  increased  zenith  distances,  and  R  the  value  of  one  turn,  then 


The  details  of  the  computation  of  the  second  and  third  terms  in  the  above  formula  are 
sufficiently  indicated  in  the  computation  shown  above.  The  first,  fourth,  and  fifth  terms  are 
explained  more  fully  on  the  following  pages  (117-119). 

Tenths  of  divisions  of  the  micrometer  head  are  usually  estimated. 

COMPUTATION   OF  APPARENT   PLACES. 

The  data  given  in  the  Boss  preliminary  general  catalogue  of  stars  for  1900  in  regard  to  a 
star,  from  which  its  apparent  place  at  the  time  of  observation  is  to  be  computed,  are  the  mean 
right  ascension  and  declination,  «m  and  8m  for  the  year  1900,  tm;  the  annual  variation  in  right 

ascension,  "Tr2;  the  annual  variation  in  declination  —5?,  (the  annual  precession  and  proper 

motion  together  constitute  the  annual  variation)  ;  and  the  secular  variation  of  the  precession 

d?d 
in  declination,  given  for  100  years,  which,  by  moving  the  decimal  point,  becomes  ~^jr-     There 

are  also  given  the  proper  motion  in  declination,  /*';  the  mean  epoch  E;  the  probable  error  of 
the  declination  at  the  mean  epoch  eaEp',  efi/j  the  probable  error  of  100  //';  and  the  probable 
error  of  the  declination  for  1910,  es.  The  probable  error  of  the  declination  for  any  date,  T,  is 


The  reduction  to  the  apparent  place  at  observation  is  made  in  two  steps;  first,  the  given 
mean  place  is  reduced  to  the  mean  place  at  the  beginning  of  the  year  of  observation,  and  upon 
that  as  a  basis  the  apparent  place  computation  is  then  made. 

Let  the  mean  right  ascension  and  declination  at  the  beginning  of  the  year  of  observation  be 
called  a0  and  80 

Then 


§0  =  §m  +  (to  -  tf  +y2(t0-  «j' 

The  Boss  catalogue   shows    that    for    the   star    4327,    <Tm  =  rt',9oo  =  16h   56m   12s,  with   an 
annual  variation  ~j^  =  -6".304.     Also  dm=dlMO  =82°    12'  07".6G.     The    annual    variation, 

7<>  s?2% 

-jf  =  -5".510,  the  secular  variation,  -~=     -".00880,  the  proper  motion,  //  =   -".001;  the 

mean  epoch,  E,  =1875.5,  and  the  probable  error,  esBp=  ±0".  03;  <v,  the  probable  error  of  100//' 
=  ±0".13,  and  the  probable  error  of  the  declination  for  1910=  ±0".05. 

i  The  correction  for  inclination  as  here  given  is  for  a  level  of  which  the  graduation  is  numbered  in  both  directions  from  the  middle.    If  the 
graduation  is  numbered  continuously  from  one  end  to  the  other  with  numbers  increasing  toward  the  objective,  the  level  correction  is 


(Compare  this  with  the  similar  formula  for  a  striding  level  on  page  23.)    If  the  numbering  on  the  level  graduation  increases  toward  the  eyepitcc  this 
formula  becomes 


DETERMINATION    OF    LATITUDE.  117 

This  star  was  observed  for  latitude  in  June,  1908,  at  St.  Anne,  111.,  Oh  43m  west  of  Washington. 

n-0  =  16h  56m  12s  -8  (68.304)  =  16h  55m  22s,  which  is  sufficiently  close  to  the  apparent  right 
ascension  for  use  in  connection  with  latitude  observations. 

£0  =  82°  12'  07".66  +  8[-5".510+K(8)(-".00880)]=82°ll'23".30.  The  probable  error 
of  the  declination  for  1908  =  V(0"-03)2  +  {  .325(0".  13)  j-  2  =  ±  0".05. 

The  apparent  declination,1  d,  at  the  instant  of  observation  may  now  be  computed  by  the 
formula  given  on  page  526  of  the  American  Ephemeris  for  1908,  namely, 

d=  d0  +  TfjL'+g  cos  (G  +  <x0)  +  h  cos  (H+a0)sin.  d0  +  icos  d0, 

in  which  g,  G,  h,  H,  and  i  are  quantities  called  independent  star  numbers  which  are  functions 
of  the  tune  only  and  are  given  in  the  Ephemeris  (pp.  532  to  539,  1908)  for  every  Washington 
mean  midnight  during  the  year,  r  is  the  elapsed  decimal  fraction  of  the  fictitious  year  and  is 
given  in  the  Ephemeris  with  the  independent  star  numbers. 

This  formula  has  been  put  in  a  more  convenient  form,  conducive  to  more  rapid  compu- 
tation, and  adapted  to  the  use  of  natural  numbers  and  Crelle's  Rechentafeln,  in  an  appendix 
to  the  Cape  Meridian  Observations,  1890-91,  entitled  "Star-Correction  Tables,"  by  W.  H. 
Finlay,  M.  A. 

The  formula  is 


in  which  /,  P'  ,  and  Q'  are  tabulated  in  the  Finlay  tables. 

P'  =  ga  cos  (G  +  a0)  and  is  tabulated  with  respect  to  the  argument  G  +  a0  and  can  be  obtained 
from  one  opening  of  the  tables  for  all  stars  and  dates. 

Q'  =  h0  cos  (H+  <TO)  sin  d0  and  is  tabulated  with  respect  to  the  arguments  (H  +  <TO)  and  d0. 

I  =  i  cos  £0  and  is  tabulated  with  respect  to  i  and  d0.  Q'  and  7  can  be  obtained  from  the 
same  opening  of  the  tables  for  any  given  star  and  date,  and  all  interpolations  involve  such 
small  tabular  differences  that  they  may  be  made  mentally. 


The  values  chosen  for  g0  and  h0  are  20".0521  and  18".500,  respectively,  so  that  x  is  generally 
negative  and  never  greater  numerically  than  unity,  while  y  is  always  positive  and  never  greater 
than  0.11;  thus  the  multiplications  by  x  and  y  can  be  easily  effected  by  Crelle's  Rechentafeln. 
x  and  y  are  functions  of  the  time  only,  and  with  sufficient  accuracy  may  usually  be  considered 
constant  for  a  single  night. 

If  the  period  over  which  the  observations  extend  on  any  night  is  not  more  than  four  hours 
long,  the  quantities  g,  7i,  G,  H,  i,  and  r  may  be  taken  from  the  Ephemeris  for  the  middle  of  the 
observing  period  and  assumed  to  be  constant  for  the  night.  The  errors  from  this  assumption 
will  be  small  and  of  both  algebraic  signs. 

The  computation  of  the  apparent  places  of  seven  stars  observed  at  the  St.  Anne  latitude 
station  is  shown  on  page  111. 

When  a  given  star  is  observed  on  several  nights  in  succession  it  is  not  necessary  to  compute 
the  apparent  place  for  every  night  of  observation.  The  apparent  place  may  be  computed 
for  certain  nights  at  intervals  of  not  more  than  three  days  and  the  declination  for  intermediate 
nights  may  be  obtained  by  interpolation. 

CORRECTION  FOR  DIFFERENTIAL   REFRACTION. 

The  difference  of  refraction  for  any  pair  of  stars  is  so  small  that  we  may  neglect  the  varia- 
tion iii  the  state  of  the  atmosphere  at  the  time  of  the  observation  from  that  mean  state  supposed 
in  the  refraction  tables,  except  for  stations  at  high  altitudes.  The  refraction  being  nearly 
proportional  to  the  tangent  of  the  zenith  distance,  the  difference  of  refraction  for  the  two  stars 
will  be  given  by 

r-r'  =  57".7sin  (z-z')  sec2z, 

1  In  the  comparatively  rare  cases  in  which  it  is  n?eessary  to  compute  the  apparent  right  ascension  of  a  star  it  may  be  done  by  the  use  of  the 
formula  given  in  Finlay's  tables. 


118 


U.   S.   COAST   AND  GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 


and  since  the  half  difference  of  zenith  distances,  as  measured  by  the  micrometer,  is  the  quantity 
applied  in  the  computation,  the  following  table  of  corrections  to  the  latitude  for  differential 
refraction  has  been  prepared  with  the  argument  one-half  difference  of  zenith  distance  at  the 
side,  and  the  argument  zenith  distance  at  the  top. 

If  the  station  is  so  far  above  sea  level  that  the  mean  barometric  pressure  at  the  station  is 
less  than  90  per  cent  of  the  mean  barometric  pressure  at  sea  level  (760mtn)  it  may  be  desirable 
to  take  this  fact  into  account  by  diminishing  the  values  given  in  the  following  table  (computed 
for  sea  level)  to  correspond  to  the  reduced  pressure.  That  is,  if  the  mean  pressure  is  10  per 
cent  less  than  at  sea  level  diminish  each  value  taken  from  the  table  by  10  per  cent  of  itself,  if  20 
per  cent  less  diminish  tabular  values  by  20  per  cent,  and  so  on.  This  need  only  be  done  roughly, 
since  the  tabular  values  are  small. 

Correction  to  latitude  for  differential  refraction  =%  (r  —  r'). 

[The  sign  of  the  correction  is  the  same  as  that  of  the  micrometer  difference.] 


One-half 
diff.of  zenith 
distances 

Zenith  distance 

0° 

10" 

20" 

25° 

30° 

35° 

40° 

45° 

0.0 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

0.5 

0.01 

0.01 

0.01 

0.01 

0.01 

0.01 

0.01 

0.02 

1.0 

0.02 

0.02 

0.02 

0.02 

0.02 

0.03 

0.03 

0.03 

1.5 

0.03 

0.03 

0.03 

0.03 

0.03 

0.04 

0.04 

0.05 

2.0 

0.03 

0.03 

0.04 

0.04 

0.04 

0.05 

0.06 

0.07 

2.5 

0.04 

0.04 

0.05 

0.05 

0.06 

0.06 

0.07 

0.08 

3.0 

0.05 

0.05 

0.06 

0.06 

0.07 

0.08 

0.09 

0.10 

3.5 

0.06 

0.06 

0.07 

0.07 

0.08 

0.09 

0.10 

0.12 

4.0 

0.07 

0.07 

0.08 

0.08 

0.09 

0.10 

0.11 

0.13 

4.5 

0.08 

0.08 

0.09 

0.09 

0.10 

0.11 

0.13 

0.15 

5.0 

0.08 

0.09 

0.10 

0.10 

0.11 

0.13 

0.14 

0.17 

5.5 

0.09 

0.10 

0.10 

0.11 

0.12 

0.14 

0.16 

0.18 

6.0 

0.10 

0.10 

0.11 

0.12 

0.13 

0.15 

0.17 

0.20 

6.5 

0.11 

0.11 

0.12 

0.13 

0.14 

0.16 

0.19 

0.22 

7.0 

0.12 

0.12 

0.13 

0.14 

0.16 

0.18 

0.20 

0.23 

7.5 

0.13 

0.13 

0.14 

0.15 

0.17 

0.19 

0.21 

0.25 

8.0 

0.13 

0.14 

0.15 

0.16 

0.18 

0.20 

0.23 

0.27 

8.5 

0.14 

0.15 

0.16 

0.17 

0.19 

0.21 

0.24 

0.  29 

9.0 

0.15 

0.16 

0.17 

0.18 

0.20 

0.23 

0.26 

0.30 

9.5 

0.16 

0.16 

0.18 

0.19 

0.21 

0.24 

0.27 

0.32 

10.0 

0.17 

0.17 

0.19 

0.20 

0.22 

0.25 

0.29 

0.34 

10.5 

0.18 

0.18 

0.20 

0.21 

0.23 

0.26 

0.30 

0.35 

11.0 

0.18 

0.19 

0.21 

0.22 

0.25 

0.28 

0.31 

0.37 

11.5 

0.19 

0.20 

0.22 

0.23 

0.26 

0.29 

0.33 

0.39 

12.0 

0.20 

0.21 

0.23 

0.25 

0.27 

0.30 

0.34 

0.40 

12.5 

0.21 

0.22 

0.24 

0.26 

0.28 

0.31 

0.36 

0.42 

13.0 

0.22 

0.22 

0.25 

0.27 

0.29 

0.33 

0.37 

0.44 

13.5 

0.23 

0.23 

0.26 

0.28 

0.30 

0.34 

0.39 

0.45 

14.0 

0.23 

0.24 

0.27 

0.29 

0.31 

0.35 

0.40 

0.47 

14.5 

0.24 

0.25 

0.28 

0.30 

0.32 

0.36 

0.41 

0.49 

15.0 

0.25 

0.26 

0.29 

0.31 

0.34 

0.38 

0.43 

0.50 

15.5 

0.26 

0.27 

0.29 

0.32 

0.35 

0.39 

0.44 

0.52 

16.0 

0.27 

0.28 

0.30 

0.33 

0.36 

0.40 

0.46 

0.54 

16.5 

0.28 

0.29 

0.31 

0.34 

0.37 

0.41 

0.47 

0.55 

17.0 

0.29 

0.29 

0.32 

0.35 

0.38 

0.43 

0.49 

0.57 

17.5 

0.29 

0.30 

0.33 

0.36 

0.39 

0.44 

0.50 

0.59 

18.0 

0.30 

0.31 

0.34 

0.37 

0.40 

0.45 

0.51 

0.  00 

18.5 

0.31 

0.  32            0.  35 

0.38 

0.41 

0.46 

0.53 

0.62 

19.0 

0.32 

0.  33            0.  36 

0.39 

0.43 

0.48 

0.54 

0.64 

19.5 

0.33 

0.  34            0.  37 

0.40 

0.44 

0.49 

0.56 

0.65 

20.0 

0.34 

0.  35    !        0.  38 

0.41 

0.45 

0.50 

0.57 

0.67 

DETERMINATION   OF   LATITUDE. 
REDUCTION   TO   THE   MERIDIAN. 


119 


If  a  star  is  observed  off  the  meridian  while  the  line  of  collimation  of  the  telescope  remains  in 
the  meridian,  the  measured  zenith  distance  is  in  error  on  account  of  the  curvature  of  the 
apparent  path  of  the  star.  Let  m  be  the  correction  to  reduce  the  measured  zenith  distance  to 
what  it  would  have  been  if  the  star  had  been  observed  upon  the  meridian. 

Then, 


in  which  T  is  the  hour-angle  of  the  star.  The  signs  are  such  that  the  correction  to  the  latitude 
(  = -Q)  is  always  plus  for  the  stars  of  positive  declination  and  minus  for  stars  of  negative  decli- 
nation (below  the  equator),  regardless  of  whether  the  star  is  to  the  northward  or  to  the  southward  of 

Tfk  77? 

the  zenith.     ^~  or  -^-  is,  then,  always  applied  as  a  correction  to  the  latitude,  with  the  sign  of  the 

right-hand  member  of  the  above  equation.  For  a  subpolar  180°—  d  must  be  substituted  for  d, 
making  the  correction  negative  in  this  case  just  as  for  stars  of  southern  declination.  The  follow- 
ing table  gives  the  corrections  to  the  latitude  computed  from  the  above  formula.  If  both  stars 
of  a  pair  are  observed  off  the  meridian,  two  such  corrections  must  be  applied  to  the  computed 
latitude. 

Correction  to  latitude  for  reduction  to  meridian. 

[Star  off  the  meridian  but  instrument  in  the  meridian.    The  sign  of  the  correction  to  the  latitude  is  positive  except  for  stars  south  of  the  equator 

and  subpolars.] 


I 

10- 

15" 

20- 

22- 

24* 

26- 

28- 

30- 

32" 

34- 

36« 

3S> 

40- 

42- 

44- 

46- 

48* 

50" 

52- 

54- 

56> 

58" 

60" 

I 

1 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.02 

.02 

89 

2 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.02 

.02 

.02 

.02 

.02 

.02 

.03 

.03 

.03 

.03 

.03 

88 

3 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

.02 

.02 

.02 

.02 

.03 

.03 

.03 

.03 

.04 

.04 

.04 

.04 

.05 

.05 

87 

4 

.01 

.01 

.01 

.01 

.01 

.02 

.02 

.02 

.02 

.03 

.03 

.03 

.04 

.04 

.04 

.05 

.05 

.06 

.06 

.06 

.07 

86 

5 

.01 

.01 

.01 

.01 

.02 

.02 

.02 

.02 

.03 

.03 

.03 

.04 

.04 

.05 

.05 

.05 

.06 

.06 

.07 

.07 

.08 

.09 

85 

6 

.01 

.01 

.01 

.02 

.02 

.02 

.03 

.03 

.03 

.04 

.04 

.05 

.05 

.06 

.06 

.07 

.07 

.08 

.08 

.09 

.10 

.10 

84 

7 

.01 

.01 

.02 

.02 

.02 

.03 

.03 

.03 

.04 

.04 

.05 

.05 

.06 

.06 

.07 

.08 

.08 

.09 

.10 

.10 

.11 

.12 

83 

8 

.01 

.02 

.02 

.02 

.03 

.03 

.03 

.04 

.04 

.05 

.05 

.06 

.07 

.07 

.08 

.09 

.09 

.10 

.11 

.12 

.13 

.14 

82 

9 

.01 

.02 

.02 

.02 

.03 

.03 

.04 

.04 

.05 

.05 

.06 

.07 

.07 

.08 

.09 

.10 

.11 

.11 

.12 

.13 

.14 

.15 

81 

10 

.01 

.02 

.02 

.03 

.03 

.04 

.04 

.05 

.05 

.08 

.07 

.07 

.08 

.09 

.10 

.11 

.12 

.13 

.14 

.15 

.16 

.17 

80 

12 

.01 

.01 

.02 

.03 

.03 

.04 

.06 

.05 

.06 

.06 

.07 

.08 

.09 

.1(1 

.11 

.12 

.13 

.14 

.15 

.16 

.17 

.19 

.20 

78 

14 

.01 

.01 

.03 

.03 

.04 

.04 

.05 

.06 

.07 

.07 

.08 

.09 

.10 

.11 

.12 

.14 

.15 

.16 

.17 

.19 

.20 

.22 

.23 

76 

16 

.01 

.02 

.03 

.03 

.04 

.05 

.06 

.07 

.07 

.08 

.09 

.10 

.12 

.13 

.14 

.15 

.17 

.18 

.20 

.21 

.23 

.24 

.26 

74 

18 

.01 

.02 

.03 

.04 

.05 

.05 

.06 

.07 

.08 

.09 

.10 

.12 

.13 

.14 

.16 

.17 

.18 

.20 

.22 

.23 

.25 

.27 

.29 

72 

20 

.01 

.02 

.04 

.04 

.05 

.06 

.07 

.08 

.09 

.10 

.11 

.13 

.14 

.1* 

.17 

.19 

.20 

.22 

.24 

.26 

.28 

.29 

.32 

70 

22 

.01 

.02 

.04 

.05 

.05 

.06 

.07 

.09 

.10 

.11 

.12 

.14 

.15 

.17 

.18 

.20 

.22 

.24 

.26 

.28 

.30 

.32 

.34 

68 

24 

.01 

.02 

.04 

.05 

.06 

.07 

.08 

.09 

.10 

.12 

.13 

.15 

.16 

.18 

.20 

.21 

.23 

.25 

.27 

.29 

.32 

.34 

.36 

66 

26 

.01 

.02 

.04 

.05 

.06 

.07 

.08 

.10 

.11 

.12 

.14 

.15 

.17 

.19 

.21 

.23 

.25 

.27 

.29 

.31 

.34 

.36 

.39 

64 

28 

.01 

.03 

.05 

.05 

.07 

.08 

.09 

.10 

.12 

.13 

.15 

.16 

.18 

.20 

.22 

.24 

.26 

.28 

.31 

.33 

.35 

.38 

.41 

62 

30 

.01 

.03 

.05 

.06 

.07 

.08 

.09 

.11 

.12 

.14 

.15 

.17 

.19 

.21 

.23 

.25 

.27 

.30 

.32 

.34 

.37 

.40 

.42 

60 

32 

.01 

.03 

.05 

.06 

.07 

.08 

.10 

.11 

.13 

.14 

.16 

.18 

.20 

.22 

.24 

.26 

.28 

.31 

.33 

..'ill 

.39 

.41 

.44 

58 

34 

.01 

.03 

.05 

.06 

.07 

.09 

.10 

.11 

.13 

.15 

.16 

.18 

.20 

.22 

.24 

.27 

.29 

.32 

.34 

.37 

.40 

.42 

.45 

56 

36 

.01 

.03 

.05 

.06 

.07 

.09 

.10 

.12 

.13 

.15 

.17 

.19 

.21 

.23 

.25 

.28 

.30 

.32 

.35 

.38 

.41 

.44 

.47 

54 

:<s 

.01 

.03 

.05 

.06 

.08 

.09 

.10 

.12 

.13 

.15 

.17 

.19 

.21 

.23 

.26 

.28 

.30 

.33 

.36 

.39 

.41 

.44 

.48 

52 

40 

.01 

.03 

.05 

.07 

.08 

.09 

.11 

.12 

.14 

.16 

.17 

.19 

.21 

.24 

.26 

.28 

.31 

.34 

.36 

.39 

.42 

.45 

.48 

5C 

45 

.01 

.03 

.05 

.07 

.08 

.09 

.11 

.12 

.14 

.16 

.18 

.20 

.22 

.24 

.26 

.29 

.31 

.34 

.37 

.40 

.43 

.46 

.49 

45 

The  catalogues  now  available  contain  so  many  stars  which  may  be  observed  for  latitude 
that  it  is  not  desirable  to  move  the  instrument  out  of  the  meridian  to  observe  a  star  which  is 
missed  as  it  crosses  the  meridian. 

COMBINATION   OF  RESULTS,   EACH   PAIR  OBSERVED  MORE  THAN   ONCE. 

Separate  values  of  the  latitude  being  computed  from  each  observation  upon  each  pair, 
it  remains  to  combine  these  in  such  a  way  as  to  obtain  the  most  probable  value  of  the  latitude 
and  to  obtain  certain  probable  errors. 


120  U.   S.   COAST  AND  GEODETIC   SURVEY  SPECIAL  PUBLICATION    NO.   14. 

Let  p  be  the  total  number  of  pairs  observed.  Let  the  number  of  observations  upon  pair 
No.  1  be  7i,,  upon  pair  No.  2,  n2,  and  so  on,  and  let  the  total  number  of  observations  at  the  sta- 
tion be  710  =  711  +  712  +  72.3  .  .  .  Let  A  be  a  residual  obtained  by  subtracting  the  result  from 
a  single  observation  on  a  certain  pair  from  the  mean  result  from  all  the  observations  upon  that 
pair.  Let  e  be  the  probable  error  of  a  single  observation  of  the  latitude,  excluding  the  error 
arising  from  defective  adopted  declinations. 

The  various  values  of  J  depend  upon  and  are  a  measure  of  the  probable  error  of  observation, 
but  are  independent  of  the  errors  of  the  adopted  declinations.  According  to  the  principles  of 
least  squares, 

0.455JJ2  0.4552"J2 


e'  = 


No.  obs.  —  No.  unknowns 


Let  g>i  be  the  mean  latitude  from  observations  on  pair  No.  1,  y>2  from  pair  No.  2,  and  so  on. 
Let  v  be  the  residual  obtained  by  subtracting  9?,,  9>2  .  .  .  in  turn  from  the  indiscriminate 
mean  for  the  station  of  <px,  <p2,  <p3  .  .  .  There  will  be  p  such  residuals,  and  they  are  a  meas- 
ure of  the  probable  error  of  the  mean  result  from  a  pair,  which  will  be  called  ep,  arising  from 
both  errors  of  observation  and  errors  of  declination. 


,       0.455  Iv2 

a*      — 

~ 


p-1 

Let  epl,  ep2     .     .     .     be  the  probable  errors,  respectively,  of  g>lt  <p.f,  <ps    .     .     .     Let  e« 
be  the  probable  error  of  the  mean  of  two  decimations.     Then 


e    i  — 


These  various  values  e2pl,  e2^,  .  .  .  differ  from  each  other  because  of  the  various 
values  of  %,  n2,  .  .  .  even  though  e2^  and  e2  are  assumed  to  be  constant,  and  the  value 
derived  above  for  e*p  is  their  mean  value.  Adding  these  various  equations,  p  in  number,  and 
taking  the  mean,  member  by  member,  there  is  obtained 


e2     e2     e2 


p 
Placing 

e 


gfi+i+i 

PL»i    "»    «i  J 


rfl       1       1  "1 

-  —  -|  —  +—  =£2 

p\_n,n,n3  J 

to  abbreviate  the  notation,  and  solving  for  e2«  there  is  obtained 


Having  determined  the  values  of  ez»  and  e2,  the  proper  relative  weights,  wlt  w2,  inversely 
proportional  to  the  squares  of  their  probable  errors,  may  now  be  assigned  to  <plt  9>2,  q>3,    .     .     . 


or 


An  exception  to  the  above  weights  arises  when  two  or  more  north  stars  are  observed  at 
one  setting  of  the  telescope  in  connection  with  the  same  south  star,  or  vice  versa,  and  the  com- 
putation is  made  as  if  two  or  more  independent  pairs  had  been  observed.  The  results  of  the 
component  pairs  in  such  a  combination  are  not  independent,  since  they  involve  in  common  the 


DETERMINATION   OP    LATITUDE.  121 

error  of  observation  and  the  error  of  declination  of  the  common  star.  The  weight  to  be  assigned 
to  each  component  pair  in  a  doublet  is  on  this  account  but  two-thirds  of  that  given  above,1 
and  to  each  component  pair  in  a  triplet  is  one-half.  The  combination  of  two  stars  on  one  side 
of  the  zenith  with  one  on  the  other  side  is  called  a  doublet,  and  three  stars  on  one  side  of  the 
zenith  with  one  on  the  other  side  is  called  a  triplet.  The  present  practice  in  the  United  States 
Coast  and  Geodetic  Survey  is  not  to  observe  doublets  or  triplets.  (See  paragraph  3  of  General 
Instructions,  p.  104.) 

If  a  combination  observed  at  one  setting  of  the  telescope  includes  two  or  more  stars  on 
each  side  of  the  zenith,  it  may  be  broken  up  in  the  computation  into  two  or  more  independent 
doublets  or  triplets,  each  of  which  may  be  treated  as  indicated  above. 

If  a  given  star  on  one  side  of  the  zenith  is  observed  in  connection  with  a  certain  star  on 
the  other  side  of  the  zenith  on  a  certain  night  (or  nights),  and  on  a  certain  other  night  (or  nights) 
is  observed  in  connection  with  some  other  star,  the  two  results  are  independent  in  so  far  as  the 
observations  are  concerned,  but  involve  a  common  adopted  declination  for  one  of  the  two 
stars  of  each  pair.  The  proper  weight  to  be  assigned  depends  in  this  case  upon  the  relative 
magnitude  of  «„  and  e,  but  is  for  their  ordinary  values  so  nearly  equal  to  the  weight  for  an 

independent  pair  that  it  may,  with  little  error,  be  assumed  to  be  such  without  going  to  the 
trouble  of  evaluating  it. 

The  weight  to  be  assigned  to  a  zenith  star  observed  in  both  positions  of  the  telescope  is 

(e2  \~l 
2e2-  +  -JT-  )    in  which  Na  is  the  number  of  nights'  observations  upon  it. 

The  most  probable  value  <p0  for  the  latitude  of  the  station  is  the  weighted  mean  of  the 
mean  results  from  the  various  pairs,  or 

_Wi< 
The  probable  error  of  <p0  is 


" 


>-l)Iw 


in  which  A<p  is  the  residual  obtained  by  subtracting  9>,,  9>2,  <p3     .  in  turn  from  <p0. 

A  concrete  illustration  of  the  processes  indicated  by  the  above  formulas  is  furnished  by 
the  following  reproduction  of  certain  parts  of  the  computation  of  the  latitude  of  the  New  Naval 
Observatory  from  observations  made  in  1897  with  a  zenith  telescope. 

1  This  may  be  made  evident  as  follows:  Let  a\  and  as  be  respectively  the  declination  plus  the  measured  zenith  distance  of  a  first  and  second 
south  star,  and  03  the  declination  minus  the  measured  zenith  distance  of  a  north  star  observed  in  combination  with  them.  Let  the  probable  errors 
of  QI,  a»,  aabeei,  ei,  e$,  respectively.  Note  that  ei,  €3,  e  s  each  include  errors  both  of  declination  and  observation.  If  the  two  component  pairs  are  com- 
puted separately  and  the  mean  taken,  the  result  is  of  the  form  f-^~^+"^^SU— •y+'j+'f  and  its  probable  error  squared  is  f-j  J  +  (^)  +  ("if)' 
Assuming  that  fi— fj— fs,  this  becomes  fai1,  the  square  of  the  probable  error  of  the  mean  result  from  the  combination.  By  the  same  reasoning  li 
may  be  shown  that  the  square  of  the  probable  error  of  the  result  from  a  single  independent  pair  is  (-rrj  +  (~^)  =id2-  The  weights  to  be  assigned 

to  the  combination  and  to  an  independent  pair  are  then  in  the  ratio  of  (|fi!)— '  and  ( jci1)— ',  or  of  j  to  1 .   If  the  weight  for  an  independent  pair  is  unity 
the  weight  of  each  component  of  a  doublet  is  therefore  two-thirds. 


122 


U.   S.   COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 


Pairs 

<t> 

Star  Nos. 

38°  55'+ 

(2058) 

09.81 

-.01 

.00 

4440 

09.80 

.00 

.00 

09.80 

4513 

08.07 

-.07 

.00 

4550 

07.92 

+.08 

.01 

08.00 

4513 

08.12 

.00 

.00 

4555 

08.13 

-.01 

.00 

08.12 

4526 

09.31 

-.34 

.12 

4550 

08.40 

+.57 

.32 

08.80 

+.17 

.03 

09.44 

-.47 

22 

09.01 

-.04 

.00 

08.85 

+.12 

.01 

08.97 

4526 

09.36 

-.26 

.07 

4555 

08.62 

+.48 

.23 

09.51 

-.41 

.  17 

08.91 

+.19 

.04 

09.  12 

-.02 

.00 

09.11 

-.01 

.00 

09.10 

Sum  

6  69 

Pair,  Star  Nos. 

B.  A.C. 

* 

V 

D« 

n 

w 

v~<;> 

i$ 

j<t> 

WJ<^ 

(lOyr.)    [c.  s.] 

(2058)        4440 

38°  55'  09".  80 

-1.00 

1.00 

2 

11 

19.80 

-0.99 

0.98 

10.78 

4513          4550 

08  .00 

+  .80 

.64 

2 

5 

0.00 

+  .81 

.66 

3.30 

4513          4555 

08  .12 

+  .68 

.46 

2 

5 

0.60 

+  .69 

.48 

2.40 

4526          4550 

08  .97 

-  .17 

.03 

6 

4* 

3.88 

-  .16 

.03 

0.12 

4526          4555 

09  .10 

-  .30 

.09 

6 

4* 

4.40 

-  .29 

.08 

0.32 

4577         (2158) 

08  .83 

-  .03 

.00 

6 

8 

6.64 

-  .02 

.00 

0.00 

(2158)         4646 

08  .72 

+  .08 

.01 

6 

8 

5.76 

+  .09 

.01 

0.08 

(2195)         4688 

09   .11 

-  .31 

.10 

5 

12 

13.32 

-  .29 

.08 

0.96 

4706          4726 

08  .25 

+  .55 

.30 

5 

12 

3.00 

+  .56 

.31 

3.72 

4742        (2233) 

08  .50 

+  .30 

.09 

5 

12 

6.00 

+  .31 

.10 

1.20 

(2254)         4847 

08  .93 

-  .13 

.02 

5 

12 

11.16 

-  .12 

.01 

0.12 

4876          4937 

08  .92 

-  .12 

.01 

5 

12 

11.04 

-  .11 

.01 

0.12 

4958         (2341) 

08  .83 

-  .03 

.00 

5 

12 

9.96 

-  .02 

.00 

0.00 

(2350)        5026 

09  .15 

-  .35 

.12 

5 

9* 

10.35 

-  .34 

.12 

1.08 

[12591       (2365) 

09  .35 

-  .55 

.30 

5 

5* 

6.75 

-  .54 

.29 

1.45 

5076          5084 

08  .64 

+  .16 

.03 

5 

12 

7.68 

+  -17 

.03 

0.36 

5115          5153 

08  .87 

-  .07 

.00 

5 

12 

10.44 

-  .06 

.00 

0.00 

5168          5178 

08  .62 

+  .18 

.03 

5 

12 

7.44 

+  .19 

.04 

0.48 

5249          5293 

08  .50 

+  .30 

.09 

5 

12 

6.00 

+  .31 

.10 

1.  20 

5313          5322 

09  .22 

-  .42 

.18 

5 

12 

14.64 

-  .41 

.17 

2.04 

5344        (2537) 

08   .44 

+  .36 

.13 

5 

12 

5.28 

+  .37 

.14 

1.68 

+3.41 

Sums 

16   .87 

-3.48 

3.63 

203 

164    .14 

31.41 

Means 

38°  55'  08".  80 

08".81 

*  For  explanation  of  these  four  weights,  see  p.  123. 


DETERMINATION   OF   LATITUDE.  123 

09 


=  0.083  -0.009  =  0.074  f 


(4.97)  =  0. 


Latitude  =  38°  55' 08".81±0".06. 

In  computing  the  values  of  w<£,  38°  55'  08".00  was  first  dropped  from  each  value  of  <f>. 

An  independent  determination  of  «»  may  be  obtained  from  the  probable  errors  of  the 
mean  declinations  of  the  stars  observed,  as  given  in  the  Boss  catalogue. 

For  the  stars  observed  at  a  station  the  mean  value  of  the  probable  error  of  the  mean  of 
two  declinations  is 

e  = 
9 

in  which  Na  is  the  total  number  of  stars  observed. 
For  a  particular  pair 

Ie\ 

/?***  ;    - 

in  which  only  the  two  stars  of  the  pair  are  included  in  the  summation  in  the  numerator.  From 
this  formula  and  from  that  given  on  page  120  (viz,  e2^=e2p  —  e2)  two  separate  values  for  e^for 
each  pair  may  be  computed.  Which  should  be  used  in  the  formula 


fixing  the  weight  to  be  assigned  to  the  mean  result  from  a  pair  ?  There  are  two  objections  to 
the  rigid  use  in  all  cases  of  the  second  value  (from  the  latitude  computation).  That  value  is 
a  mean  for  all  the  pairs  of  a  list,  and  in  using  it  the  fact  that  some  declinations  have  very  much 
larger  probable  errors  than  others  in  the  same  list  is  ignored.  Moreover,  in  practice,  the  formula 
e2^  =  e2p  —  s2  is  sometimes  found  to  give  a  value  for  e^  which  is  so  small  as  to  be  evidently  erro- 
neous, and  sometimes  e2^  is  even  negative,  which  is  an  absurdity.  On  the  other  hand,  when- 

2ez 
ever  the  value  e2^  =  -^fis  smaller  than  e2^  =  e2  —  s2p,  and  that  is  usually  the  case,  it  indicates 

that  there  is  in  the  observations  some  error  peculiar  to  each  star,  which  combines  with  the 
declination  error,  and  so  apparently  increases  it.  When  such  errors  exist,  the  weights  should 
be  correspondingly  reduced,  and  therefore  the  values  of  «2«  =  e2p— s2  should  be  used  in  the 
weighting. 

The  following  method  of  weighting,  therefore,  seems  to  be  the  best  for  use  in  the  office 

(e2  \~' 
e\  4.  —  )    ,  use  for  each  pair  the  larger 
Wn/ 

Ie2 

of  the  two  available  values  of  e2^,  namely,  e2%  =  — j-*  and  e2^  =  e2J>  —  s2.  By  so  doing  all  the  dis- 
advantages of  each  of  the  two  methods  discussed  in  the  preceding  paragraph  are  avoided.  To 
find  quickly  which  of  the  values  of  e2**  from  the  mean  place  computation  are  greater  than  e2»  = 
e2p  —  s2  one  may  first  note  on  the  list  of  mean  places  for  what  stars  e2t  exceeds  2  (e2p  —  s2).  Only 
pairs  involving  such  stars  need  be  examined  further.  To  illustrate,  of  the  pairs  involved  in  the 
latitude  computation  shown  on  page  122,  there  were  only  four  for  which  the  mean  place  com- 
putation gave  values  of  e2»  exceeding  0.074.  The  stars  involved  in  these  four  pairs  were  4526, 
4550,  4555,  (2350),  5026,  [1259],  (2365),  and  the  corresponding  values  of  e2t  were  0.37,  0.08,  0.10, 

2e2 
0.18,  0.24,  0.08,  0.73.     The  weights  assigned  to  these  four  pairs  therefore  depend  upon  e2f  =  — j-1 

in  each  case. 


124  U.   S.    COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 

COMBINATION   OF   RESULTS  WHEN   EACH   PAIR   IS   OBSERVED   BUT   ONCE. 

It  is  the  present  practice  of  this  Survey  to  observe  a  pair  of  stars  only  once  at  a  station, 
and  in  the  final  computations  the  resulting  latitude  from  each  pair  observed  is  given  unit  weight. 
(See  the  first  paragraph  under  the  heading  "General  Notes  on  Computations  of  Latitude  in 
the  U.  S.  Coast  and  Geodetic  Survey"  on  p.  115.) 

Whenever  the  plan  of  observing  each  pair  but  once  at  a  station  is  carried  out  the  method  of 
combining  results  and  computing  probable  errors  outlined  in  the  preceding  pages  fails,  and  for 
it  must  be  substituted  the  following  procedure,  for  which  little  additional  explanation  is  needed: 

2  _  0.455  2V 

in  which  ep  is  the  probable  error  of  the  result  from  a  pah-,  including  both  the  error  of  observation 
and  the  declination  errors,  v  is  the  residual  obtained  by  substracting  the  latitude  from  a  single 
pair  from  the  indiscriminate  mean  of  all  the  pairs,  and  p  is  the  number  of  pairs.  In  the  field 
computation  and  also  in  the  final  computation  this  indiscriminate  mean  is  considered  to  be  the 
final  value  of  the  latitude.  Its  probable  error  is 


0.455  2V 

>(p-\) 

No  value  of  the  probable  error  of  observation  not  involving  the  decimation  error  is  available 
from  such  a  field  computation.  But  the  computed  values  of  ep  and  e<t>  give  sufficiently  good 
indications  of  the  accuracy  of  the  observations  to  enable  the  observer  to  decide  in  the  field 
whether  the  instrument  is  in  good  condition  and  whether  more  observations  are  needed  and 
that  is  all  that  is  necessary.  (See  p.  104.) 

If  desired,  the  office  computation  may  be  carried  further  as  the  probable  error  of  the  decima- 
tion of  a  star  e*  may  be  obtained  from  the  catalogue. 

2$ 
The  probable  error  of  a  single  observation  is  given  by  the  formula  e*  =  e2p  — «-»?,  in  which 

N,  is  the  total  number  of  stars  observed. 

If  weights  were  given  each  pan*  (not  the  present  practice  in  this  Survey),  the  weight  to  be 
assigned  to  a  pan-  would  be 


e 

in  which  for  each  pair  e2   —— TJ  the  summation  covering  the  two  stars  of  that  pan-  only. 

™       * 

DETERMINATION   OF  LEVEL  AND  MICROMETER  VALUES. 

For  methods  of  determining  the  level  value  see  page  46. 

Until  recently  the  method  most  frequently  used  in  this  Survey  for  determining  the  microm- 
eter value  is  as  follows:1  The  tune  is  observed  that  is  required  for  a  close  circumpolar  star, 
near  elongation,  to  pass  over  the  angular  interval  measured  by  the  screw.  Near  elongation  the 
apparent  motion  of  the  star  is  nearly  vertical  and  nearly  uniform.  That  one  of  the  four  close 
circumpolars  given  in  the  Ephemeris,  namely,  a,  d,  and  A  Ursae  Minoris  and  51  Cephei,  may  be 
selected  which  reaches  elongation  at  the  most  convenient  hour.  In  selecting  the  star  it  may  be 
assumed  with  sufficient  accuracy  that  the  elongations  occur  when  the  hour-angle  is  six  hours 
on  either  side  of  the  meridian.  In  planning  the  observations  and  in  making  the  computation 
it  is  necessary  to  know  the  tune  of  elongation  more  accurately,  and  it  may  be  computed  from 
the  formula 

cos  t-E  =  cot  d  tan  <£ 

1  See  Appendix  No.  3,  United  States  Coast  and  Geodetic  Survey,  Report  for  19(10,  for  a  full  discussion  of  the  determination  of  micrometer 
value. 


DETERMINATION    OF    LATITUDE.  125 

Chronometer  time  of  elongation  =ct  —  4T±tE,  the  plus  sign  being  used  for  western  elonga- 
tion and  the  minus  for  eastern  elongation.  tK  is  the  hour-angle  at  elongation  reckoned  eastward 
or  westward  from  upper  culmination,  and  AT  is  the  chronometer  correction. 

If  desired  £E,  the  zenith  distance  of  the  star  at  elongation  may  be  computed  from  the 
formula 

cos  £E  =  cosec  d  sin  $ 

It  is  advisable  to  have  the  middle  of  the  series  of  observations  about  elongation.  The 
observer  may  obtain  an  approximate  estimate  of  the  rate  at  which  the  star  moves  along  the 
micrometer  by  a  rough  observation  or  from  previous  record,  and  time  the  beginning  of  his 
observations  accordingly. 

To  begin  observations  the  star  is  brought  into  the  field  of  the  telescope  and  to  the  proper 
position,  the  telescope  is  clamped  both  in  zenith  distance  and  azimuth,  the  micrometer  is  made 
to  read  an  integral  number  of  turns,  and  the  bubble  is  brought  approximately  to  the  middle 
of  the  level  tube.  The  chronometer  time  of  transit  of  the  star  across  the  thread  is  observed 
and  the  level  read.  The  micrometer  thread  is  then  moved  one  whole  turn  in  the  direction  of  the 
apparent  motion  of  the  star,  the  tune  of  transit  again  observed  and  the  level  read,  and  the 
process  repeated  until  a  sufficiently  large  portion  of  the  middle  of  the  screw  has  been  covered 
by  the  observations  to  correspond  with  what  is  actually  used  in  the  latitude  observations.  If 
desired,  an  observation  may  be  made  at  every  half  turn,  or  even  at  every  quarter  turn,  by 
allowing  an  assistant  to  read  the  level.  It  is  well  to  note  the  temperature. 

The  form  of  record  and  computation  is  shown  below,  the  first  four  columns  being  the 
record,  and  the  remainder  the  computation,  of  the  value  of  one  turn  of  micrometer  from  observa- 
tions made  at  the  New  Naval  Observatory  June  18,  1897. 

•£  =  38°  55'  08".S. 

For  the  star  B.  A.  C.  8213  at  the  time  of  observation  <x  =  23h  27™  458.6  and  5  =  86°  44' 
13". 4.  The  chronometer  correction  at  the  time  of  the  observations  was  known  to  be  +  28.3. 
Whence  the  chronometer  time  of  eastern  elongation  was  computed  to  be  17h  38m  168.5  and  the 
zenith  distance  51°  00'. 5. 


126 


U.    S.    COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 


Computation  of  value  of  micrometer. 

Station  New  Naval  Observatory,  Washington,  D.  C.    Observer,  O.  B.  F.    Star,  B.  A.  C.  8213  E.  E.    Date,  June  18, 1897.    Instrument,  Zenith 

telescope,  No.  4.] 


Mi- 
crome- 
ter 
read- 
ing 

Chronom- 
eter time 
of  observa- 
tion 

Level 

Time 
from 
elonga- 
tion 

Reduc- 
tion to 
mean 
state  o 
level 

Corrections 

Reduced 
time 

Time  at  20 
turns 

J 

J' 

n 

s 

Time 

Level 

t 

Am     s 

d 

d 

m 

t 

s 

s 

A  m     s 

Am     s 

s 

s 

35 

17  15  08.5 

|13.  1 
168.4 

39.9 
101.9 

)„ 

—0.10 

+2.3 

-0.1 

17  15  10.7 

17  28  10.7 

+4.7 

-0.3 

34 

16  02.0 

22.2 

+2.1 

-0.1 

16  Ot.O 

12.0 

+3.4 

-1.2 

33 

16  53.5 

21.4 

+  1.9 

-0.1 

16  55.3 

11.3 

+  4.1 

-0.2 

32 

17  45.  0 

T13.2 
168.6 

40.0 
102.0 

I     20.5 

+0.15 

+  1.6 

+0.1 

17  46.  7 

10.7 

+4.7 

+0.7 

31 

18  37.5 

19.6 

+  1.4 

+0.1 

18  39.  0 

11.0 

+4.4 

+0.7 

30 

19  30.  0 

18.8 

+  1.3 

+0.1 

19  31.4 

11.4 

+4.0 

+  0.7 

29 

20  22.5 

17.9 

+  1.1 

+0.1 

20  23.7 

11.7 

+3.7 

+0.7 

28 

21  16.0 

17.0 

+0.9 

+0.1 

21  17.0 

13.0 

+2.4 

-0.3 

27 

22  07.5 

16.1 

+0.8 

+0.1 

22  08.4 

12.4 

+3.0 

+0.7 

26 

23  00.0 

15.2 

+0.7 

+0.1 

23  00.8 

12.8 

+  2.6 

+0.6 

25 

23  53.0 

14.4 

+  0.6 

+0.1 

23  53.7 

13.7 

+  1.7 

0.0 

24 

24  45.5 

13.5 

+0.5 

+0.1 

24  46.  1 

14.1 

+  1.3 

0.0 

23 

25  37.5 

12.6 

+0.4 

+0.1 

25  38.0 

14.0 

+  1.4 

+  0.4 

22 

26  30.5 

11.8 

+0.3 

+  0.1 

26  30.  9 

14.9 

+0.5 

-0.2 

21 

2V  23.0 

[13.2 
[68.3 

40.0 
101.8 

'     10.9 

-0.10 

+0.2 

-0.1 

27  23.1 

15.1 

+0.3 

0.0 

20 

28  16.0 

10.0 

+0.2 

-0.1 

28  16.1 

16.1 

-0.7 

-0.7 

'13.2 

40.  0 

19 

29  08.0 

9.1 

+0.15 

+0.1 

+0.1 

29  08.  2 

16.2 

-0.8 

-0.5 

k68.  5 

102.1 

18 

3000.5 

8.2 

+0.1 

+0.1 

30  00.7 

16.7 

-1.3 

-0.6 

17 

30  53.  0 

7.4 

+0.1 

+0.1 

30  53.  2 

17.2 

-1.8 

-0.8 

16 

31  44.5 

6.5 

+0.1 

+0.1 

31  44.  7 

16.7 

-1.3 

0.0 

15 

32  37.  0 

5.7 

0.0 

+0.1 

32  37.  1 

17.1 

-1.7 

0.0 

14 

33  29.5 

4.8 

0.0 

+0.1 

33  29.6 

17.6 

-2.2 

-0.2 

13.2 

40.  1 

13 

34  22.0 

68.3 

101.9 

3.9 

0.00 

0.0 

0.0 

34  22.0 

18.0 

-2.6 

-0.3 

12 

35  14.0 

3.0 

0.0 

0.0 

35  14.0 

18.0 

-2.6 

+  0.1 

11 

36  06.5 

2.2 

0.0 

0.0 

36  08.5 

18.5 

-3.1 

-0.1 

10 

36  58.  <i 

1.3 

0.0 

0.0 

36  58.  5 

18.5 

-3.1 

+0.2 

9 

37  50.5 

0.4 

0.0 

0.0 

3750.5 

18.5 

-3.1 

+  0.6 

8 

38  43.5 

-0.4 

0.0 

0.0 

38  43.  5 

19.5 

-4.1 

-0.1 

7 

39  35.5 

-1.3 

0.0 

0.0 

39  35.  5 

19.5 

-4.1 

+0.2 

6 

40  28.0 

-2.2 

0.0 

0.0 

40  28.  0 

20.0 

-4.6 

0.0 

5 

41  19.5 

-3.0 

0.0 

0.0 

41  19.5 

19.5 

-4.1 

+0.9 

Mean 

17  28  15.4 

Assumed  value  of  Rl  =  52s. 


2480  r, 
r, 


log  15 
log  cos  d 


=  +  820.3 
=  +     0.3308 
528.3308 

=         1.7187573 

1.1760913 

=          8.7552522 


1.6501008 
44".679 
Corr.  for  refraction  —   0    .030 


One  turn  44".649 

For  explanation  of  notation,  see  page  128. 


DETERMINATION    OF    LATITUDE. 


127 


Because  of  the  curvature  of  the  apparent  path  of  the  star  its  rate  of  change  of  zenith  distance 
is  not  constant,  even  near  elongation.  The  rate  of  change  at  elongation  may  readily  be  com- 
puted. It  is  at  that  instant  in  seconds  of  arc  15  cos  d  per  second  of  sidereal  time.  The  table 
of  curvature  corrections  given  below  enables  one  to  correct  the  observed  times  to  what  they 
would  have  been  if  in  the  place  of  the  actual  star  there  were  substituted  an  ideal  star  whose 
motion  was  vertical  at  a  constant  rate  15  cos  d  and  which  coincided  with  the  actual  star  at 
the  instant  of  elongation. 

Correction  for  curvature  of  apparent  path  of  star,  in  computation  of  micrometer  value. 

[The  correction  tabulated  is  -  (15  sin  I")2 13—  y™  (15  sin  I")1 t5  in  which  t  is  the  time  from  elongation.    Apply  the  corrections  given  in  the 
table  to  the  observed  chronometer  times,  adding  them  before  either  elongation,  and  subtracting  them  after  either  elongation.] 


T 

Corr. 

T 

Corr. 

T 

Corr. 

T 

Corr. 

T 

Corr. 

T 

Corr. 

m 

» 

m 

s 

m 

s 

m 

s 

m 

« 

m 

» 

6 

0.0 

16 

0.8 

26 

3.3 

36 

8.9 

46 

18.5 

56 

33.3 

7 

0.1 

17 

0.9 

27 

3.7 

37 

9.6 

47 

19.7 

57 

35.1 

8 

0.1 

18 

1.1 

28 

4.2 

38 

10.4 

48 

21.0 

58 

37.0 

9 

0.1 

19 

1.3 

29 

4.6 

39 

11.3 

49 

22.3 

59 

39.0 

10 

0.2 

20 

1.5 

30 

5.1 

40 

12.2 

50 

23.7 

60 

41.0 

11 

0.2 

21 

1.8 

31 

5.7 

41 

13.1 

51 

25.2 

61 

43.1 

12 

0.3 

22 

2.0 

32 

6.2 

42 

14.1 

52 

26.7 

62 

45.2 

13 

0.4 

23 

2.3 

33 

6.8 

43 

15.1 

53 

28.3 

63 

47.4 

14 

0.5 

24 

2.6 

34 

7.5 

44 

16.2 

54 

29.9 

64 

49.7 

15 

0.6 

25 

3.0 

35 

8.2 

45 

17.3 

55 

31.6 

65 

52.1 

In  the  computation  the  fifth  column  shows  the  values  of  T,  and  the  seventh  column  the 
resulting  curvature  corrections. 

When  the  reading  of  the  level  changes,  it  indicates,  upon  the  usual  assumption  that  the 
relation  between  the  level  vial  and  the  telescope  remains  constant,  that  the  inclination  of  the 
telescope  has  changed.  The  effect  of  the  movement  of  the  telescope  may  be  eliminated  in  the 
computation  by  applying  to  each  observed  time  the  correction  in  seconds  of  time, 

±  {(»-«)-(»'-*')} 


30  cos  d 

to  reduce  it  to  what  it  would  have  been  if  the  readings  of  the  north  and  south  end  of  the  bubble 
had  been  n'  and  s',  respectively. 

If,  as  in  the  present  case,  the  level  graduation  is  numbered  continuously  from  one  end  to 
the  other  with  the  numbers  increasing  toward  the  eye  end,  instead  of  being  numbered  in  both 
directions  from  the  middle,  the  required  correction  becomes 

d 


'30  cos  d 

In  each  of  these  formula?  the  plus  sign  is  to  be  used  for  western  elongation  and  the  minus 
sign  for  eastern  elongation.  It  is  convenient  to  take  for  the  assumed  n'  and  s'  the  actual 
readings  at  some  one  moment  during  the  set  of  observations. 

Zenith  telescope  No.  4  had  two  latitude  levels,  and  the  correction  was  computed  by  taking 
the  mean  of  the  two  and  using  the  mean  value  of  d  (=  1".482).  The  sixth  column  shows  the 
mean  values  of  (n' +  s')  — (n  +  s)  and  the  eighth  column  the  resulting  corrections,  the  factor 

on  d     .  being  0.  87. 
30  cos  d 

Let  RI  be  an  assumed  approximate  value  of  one  turn  in  time  and  let  rl  be  a  required  cor- 
rection to  .Bj.  Let  jT0  be  an  approximate  value  of  the  chronometer  time  of  transit  of  the  star 
across  the  micrometer  line  set  at  20  turns  (the  middle  of  the  screw)  and  t0  a  required  correction 
to  T0.  Then,  upon  the  assumption  that  the  screw  has  a  uniform  value  throughout  the  part 


128  U.   S.   COAST  AND  GEODETIC   SURVEY   SPECIAL  PUBLICATION    NO.   14. 

observed  upon  and  that  the  star  moves  in  the  direction  of  increasing  readings  (western  elongation)  , 
for  each  observed  time  an  observation  equation  may  be  written  of  the  form 

t+(20-R0]  (R1  +  r1}-(Tl,+t,}=0 

in  which  t  is  the  observed  time  of  transit  across  the  line  set  at  the  reading  /?„  after  correction 
for  curvature  and  level.     After  transposition  this  may  be  written 

(20-/?0)r1-^0  =  J 
in  which 

J-T0-p+(20-12(,)JBJ 

whence  the  normal  equations  become 

1(20-  R.V-r,  -2(20-  R0}t0  =  2(20-  R0}d 
=  -  IJ. 


If  the  turns  observed  upon  are  symmetrical  about  20,  1(20  —  R0)  becomes  zero.  If,  more- 
over, as  in  the  numerical  case  here  shown,  T0  is  purposely  taken  equal  to  the  mean  value  of 
t+  (20  —  R0)Rlt  2  A  is  zero  and  t0  derived  from  the  second  normal  equation  is  necessarily  zero. 
Also  the  first  normal  equation  reduces  to  the  working  form 


If  the  star  is  observed  at  eastern  elongation  it  moves  in  the  direction  indicated  by  decreasing 
micrometer  readings  and  throughout  the  preceding  formulae  R,,  —  20  must  be  substituted  for 
20  -R0. 

In  the  computation  form  printed  above,  the  values  of  t  +  (R0  —  20)  R{  are  shown  in  the  column 
headed  "Time  at  20  turns,"  Rl  being  assumed  =  52s.  T0  was  assumed=  17h  28m  15.84,  the  mean 
for  this  column,  and  the  J's  written  accordingly. 

The  equation  2(R9-20)2r1  =  2(R9-20)J  reduces  numerically  to  2480r11  =  820.3. 

A'  is  the  residual  obtained  by  substituting  the  derived  value  rt  in  each  observation  equation, 
or  J'-J-(B0-20)rt. 

The  remainder  of  the  computation  needs  no  explanation  except  that  the  correction  for  refrac- 
tion to  be  applied  to  the  value  of  one  turn  is  the  change  of  refraction  for  a  change  of  zenith 
distance  equal  to  one  turn,  or  in  the  most  convenient  form  for  use,  it  is  the  value  of  one  turn  in 
minutes  of  arc  times  the  difference  of  refraction  for  1'  at  the  altitude  at  which  the  star  was 
observed  (approximately  =<j&).  The  difference  of  refraction  for  1'  may  be  obtained  from  any 
table  of  mean  refractions  with  sufficient  accuracy.  The  correction  for  refraction  is  always 
negative,  since  the  change  of  refraction  is  always  such  as  to  make  a  star  appear  to  move  slower 
than  it  really  does. 

It  will  sometimes  be  necessary  to  apply  a  correction  for  rate.  This  correction,  to  be  applied 
to  the  computed  value  of  one  turn,  is  in  seconds  of  arc 

(rate  of  chronometer  in  seconds  per  day)  (value  of  one  turn  in  seconds  of  arc) 

86400" 

The  correction  is  negative  if  the  chronometer  runs  too  fast. 

The  micrometer  value  is  sometimes  determined  by  turning  the  micrometer  box  90°  and 
observing  upon  a  close  circumpolar  near  culmination.  There  are  two  serious  objections  to  this 

'  In  this  computation  it  becomes  necessary  to  find  the  sum  ol  the  series  l»+2*+3!+4*    ....    +15*.    It  is  convenient  for  this  purpose  to 
use  the  ,'ormula  ls+21+3'+4!    .    .    .    +i*—  3  +  2+5-    Occasionally  in  least  square  computations  it  becomes  necessary  to  compute  the  sum  of  a 


similar  series  of  fourth  powers.    One  may  then  use  the  formula  l'+2<+3'+4<    .    .    +i>_++_£.   To  obtain  the  sum  of  the  series  (J)<+(J)'+ 

(»)'+(l)(+(  i)«    .    .    .     +#,  apply  the  formula  to  the  series  l<+2<+3<+4<    .    .    +(4i)<  and  divide  by  256-  4«.    See  Sammlung  von  Formilndtr 
reinfn  und  a.igewandfen  Afathematik  von  Dr.  W.  Laska,  p.  88  (Braunschweig,  1S88-1S94). 


DETERMINATION   OF    LATITUDE.  129 

procedure.  The  focal  adjustment  is  liable  to  be  disturbed  more  or  less  when  the  micrometer 
box  is  turned,  and  a  corresponding  constant  error  introduced  into  the  result.  In  observing 
at  elongation  the  telescope  is  depended  upon  to  be  stable  in  zenith  distance,  the  direction  in 
which  it  is  designed  to  be  stable,  and  the  level  readings  furnish  a  means  of  correcting  in  large 
p<irt  for  small  movements  in  that  direction.  But  when  the  observations  are  made  at  culmination 
the  instrument  is  depended  upon  to  remain  fixed  in  azimuth,  the  direction  in  which,  because  of 
its  peculiar  design,  it  is  weakest,  and  there  is  no  check  upon  changes  in  azimuth  corresponding 
to  the  level  readings.  Hence,  it  is  not  advisable  to  observe  for  micrometer  value  at  culmination. 
The  only  modifications  in  the  computations  are  that  there  are  no  corrections  for  level  or 
refraction,  and  that  in  computing  the  curvature  correction  r  is  now  the  hour-angle.  The 
curvature  correction  is  additive  before  either  culmination,  and  subtractive  after  it. 

It  is  decidedly  questionable  whether  it  is  advisable  to  determine  the  mean  value  of  the 
micrometer  screw  by  observations  upon  close  circumpolars  either  at  culmination  or  elongation. 
Such  observations  consume  a  great  deal  of  time  both  in  observation  and  in  the  subsequent 
computation,  and  experience  shows  that  they  are  subject  to  unexpectedly  large  and  unexplained 
errors.  For  example,  during  the  observations  for  variation  of  latitude  at  Waikiki,  Hawaiian 
Islands,  in  1891-92,  the  micrometer  value  was  thus  determined  twelve  times.  The  results 
show  a  range  of  about  0".13  or  one  three-hundred-and-thirtieth  of  the  mean  value,  corresponding 
to  a  range  of  about  3.3  millimeters  in  the  distance  between  the  objective  and  the  micrometer 
line,  though  the  draw  tube  was  kept  clamped  continuously,  and  the  range  of  temperature  during 
the  entire  year  was  only  about  11°  C.  (Coast  and  Geodetic  Survey  Keport,  1892,  Part  II,  p.  61.) 
Similaily,  sixteen  determinations  of  the  value  of  a  micrometer  used  at  fifteen  stations  on  the 
Mexican  Boundary  Survey  of  1892-93  showed  a  range  of  0".33  or  one  one-hundred-and-ninetieth 
of  the  mean  value.1  In  this  case  the  draw  tube  was  undamped  and  the  telescope  refocused 
at  the  beginning  of  the  observations  at  each  station.  The  observed  value  was  apparently  not  a 
function  of  the  temperature.  The  San  Francisco  series  of  observations  for  variation  of  latitude 
also  show  a  similar  large  range  in  the  observed  micrometer  value  (viz:  0".17).  (Coast  and  Geo- 
detic Survey  Report,  1893,  Part  II,  p.  447.)  In  general,  whenever  the  micrometer  value  is 
determined  repeatedly  by  the  circumpolar  method  so  large  a  range  of  results  is  developed  as  to 
force  one  to  suspect  that  large  constant  errors  are  inherent  in  this  method  of  observation.  It 
can.  hardly  be  urged  that  the  differences  between  the  results  represent  actual  changes  in  the 
micrometer  value,  for  such  differences  are  developed  even  when  successive  determinations  are 
made  during  a  single  evening.  Moreover,  whenever  the  mean  micrometer  value  is  determined 
from  the  latitude  observations  themselves  it  is  frequently  found  to  differ  radically  from  that 
derived  from  circumpolar  observations  on  the  same  nights.  So  marked  and  so  frequent  has  the 
latter  form  of  disagreement  been,  that  many  of  the  office  latitude  computations  have  actually 
been  made  during  the  last  few  years  by  rejecting  the  micrometer  value  from  circumpolar  observa- 
tions, when  there  is  a  marked  difference  between  it  and  the  value  computed  from  the  latitude 
observations  as  indicated  below,  and  using  the  latter  value  in  the  latitude  computation. 

DETERMINATION  OF  MICROMETER  VALUE  FROM  LATITUDE  OBSERVATIONS. 

After  considering  the  above  facts  and  conclusions  this  Survey  decided  to  adopt  the  method 
of  computing  the  micrometer  value  from  the  latitude  observations,  and  since  the  beginning  of 
the  year  1905  no  observations  have  been  made  on  close  circumpolar  stars  for  that  purpose. 

The  total  range  in  the  values  of  one  turn  of  the  micrometer  screw  of  zenith  telescope  No. 
2,  as  determined  from  the  latitude  observations  for  36  of  the  63  stations  established  by  Assistant 
W.  H.  Burger,  from  1905  to  1908,  is  0".17.  This  is  one  two  hundred  and  seventy-third  of  the 
mean  value. 

As  to  the  accuracy  of  the  micrometer  value  determin'ed  from  the  latitude  observations, 
it  may  be  noted  that  if  it  be  assumed  that  the  probable  error  of  a  single  observation  of  latitude 

1  Report  of  the  International  Boundary  Commission,  United  States  and  Mexico,  1891-1896  (Washington,  1898),  p.  103. 
8136°— 13 9 


130  U.   S.    COAST   AND   GEODETIC   SUEVEY   SPECIAL   PUBLICATION    NO.   14. 

is  ±0".40,  of  the  mean  of  two  declinations  is  ±0".16  (see  p.  133)  and  of  the  latitude  as 
derived  from  independent  pairs  is  ±0".10,  the  probable  error  of  the  micrometer  value,  as 
determined  from  a  single  observation  upon  a  pair  having  a  difference  of  zenith  distance  of  ten 
turns  would  be 


'.40)2  +  4(0.16)2+(0.10)2  =  ±0".05. 

There  can  be  little  doubt,  therefore,  that  the  mean  micrometer  value  determined  from 
all  the  latitude  observations  at  a  station  is  more  accurate  than  that  determined  from  even 
three  or  four  sets  of  circumpolar  observations  each  requiring  an  hour  or  more  of  time. 

It  has  been  urged  that  to  determine  an  instrumental  constant  from  the  observations  in 
the  computation  of  which  it  is  to  be  used  is  a  questionable  procedure;  that  it  "smooths  out" 
the  results,  but  probably  does  not  give  real  accuracy.  The  force  of  this  objection  disappears 
when  one  contrasts  the  proposed  practice  of  deriving  a  single  instrumental  constant  from  ob- 
servations on  twelve  or  more  pairs  with  the  usual  and  unquestioned  practice  in  transit  time 
computations  of  deriving  three  instrumental  constants  (two  azimuth  and  one  collimation  con- 
stant) from  only  ten  to  twelve  observations  on  as  many  stars. 

It  should  be  noted  that  the  form  of  the  computation  of  circumpolar  micrometer  obser- 
vations given  on  page  126  is  especially  adapted  to  the  detection  of  irregularities  and  periodic 
errors,  as  they  will  at  once  become  evident  from  an  inspection  of  the  values  of  J'.  One  com- 
mon form  of  irregularity  in  screws  is  a  continuous  increase  in  the  value  from  one  end  to  the 
other,  in  which  case  J'  tends  to  have  the  same  sign  at  the  two  ends  of  the  set  and  the  opposite 
sign  in  the  middle. 

To  derive  the  mean  micrometer  value  from  the  latitude  observations  let  M^  be  the  differ- 
ence, in  turns,  of  the  micrometer  readings  on  the  two  stars  of  a  pair,  taken  with  the  same  sign 
as  in  the  latitude  computation,  let  r,  be  the  required  correction  to  the  assumed  value  of  one-half 
turn  with  which  the  computation  of  the  latitude  was  made,  let  p  be  the  number  of  pairs,  and 
let  c  be  the  correction  to  the  mean  latitude  <f>0.  Let  J<£  have  the  same  meaning  as  before, 
viz,  0o~0u  <f>o~<j>2>  etc.  (See  computation  on  p.  114.)  For  each  pair  an  observation  equation 
of  the  form  c  —  M^r^  +  A<j>  =  0  may  be  written.  The  resulting  normal  equations,  from  which  rl 
may  be  derived,  are 

—  2  Jf,c  +  ^  M2!?1!  —  I  M^(j)  =  0 

The  computation  will  be  sufficiently  accurate  if  M^  is  carried  to  tenths  of  turns  only,  and 
as  here  indicated  without  assigning  weights  to  the  separate  pairs. 

To  the  preliminary  values  of  <£„  <j>2  .  .  .  ,  the  results  from  the  separate  pairs,  may 
now  be  applied  the  corrections  M^  and  the  latitude  computation  completed  as  before. 

REDUCTION  TO  SEA  LEVEL. 

The  reduction  of  the  observed  latitude  to  sea  level  is  given  by  the  expression 

J0=- 0.000171  h  sin  2<f> 

in  which  J<j>  is  the  correction  in  seconds  of  arc  to  be  applied  to  the  observed  latitude,  h  is  the 
elevation  of  the  station  above  sea  level  in  meters,  and  <£  is  the  latitude  of  the  station.  This 
correction  may  be  gotten  from  the  following  table: 


DETERMINATION    OF    LATITUDE. 

Reduction  of  latitude  to  sea  level. 

[The  correction  is  negative  in  every  case.] 


131 


* 

ft 

5° 
85° 

10" 
80° 

18° 
75° 

20° 
70° 

25° 
65° 

30° 
60° 

35° 
55° 

40° 
50° 

45° 

Feet 

Meiers 

// 

// 

// 

// 

// 

// 

// 

/' 

// 

100 

30 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

0.01 

0.01 

200 

61 

.00 

.00 

.01 

.01 

.01 

.01 

.01 

.01 

.01 

300 

91 

.00 

.01 

.01 

.01 

.01 

.01 

.01 

.02 

.02 

400 

122 

.00 

.01 

.01 

.01 

.02 

.02 

.02 

.02 

.02 

500 

152 

.00 

.01 

.01 

.02 

.02 

.02 

.02 

.03 

.03 

600 

183 

.01 

.01 

.02 

.02 

.02 

.03 

.03 

.03 

.03 

700 

213 

.01 

.01 

.02 

.02 

.03 

.03 

.03 

.04 

.04 

800 

244 

.01 

.01 

.02 

.03 

.03 

.04 

.04 

.04 

.04 

900 

274 

.01 

.02 

.02 

.03 

.04 

.04 

.04 

.05 

.05 

1000 

305 

.01 

.02 

.03 

.03 

.04 

.05 

.05 

.05 

.05 

1100 

335 

.01 

.02 

.03 

.04 

.04 

.05 

.05 

.06 

.06 

1200 

366 

.01 

.02 

.03 

.04 

.05 

.05 

.06 

.06 

.06 

1300 

396 

.01 

.02 

.03 

.04 

.05 

.06 

.06 

.07 

.07 

1400 

427 

.01 

.02 

.04 

.05 

.06 

.06 

.07 

.07 

.07 

1500 

457 

.01 

.03 

.04 

.05 

.06 

.07 

.07 

.08 

.08 

1600 

488 

.01 

.03 

.04 

.05 

.06 

.07 

.08 

.08 

.08 

1700 

518 

.02 

.03 

.04 

.06 

.07 

.08 

.08 

.09 

.09 

1800 

549 

.02 

.03 

.05 

.06 

.07 

.08 

.09 

.09 

.09 

1900 

579 

.02 

.03 

.05 

.06 

.08 

.09 

.09 

.10 

.10 

2000 

610 

.02 

.04 

.05 

.07 

.08 

.09 

.10 

.10 

.10 

2100 

640 

.02 

.04 

.05 

.07 

.08 

.09 

.10 

.11 

.11 

2200 

671 

.02 

.04 

.06 

.07 

.09 

.10 

.11 

.11 

.11 

2300 

701 

.02 

.04 

.06 

.08 

.09 

.10 

.11 

.12 

.12 

2400 

732 

.02 

.04 

.06 

.08 

.10 

.11 

.12 

.12 

.13 

2500 

762 

.02 

.04 

.07 

.08 

.10 

.11 

.12 

.13 

.13 

2600 

792 

.02 

.05 

.07 

.09 

.10 

.12 

.13 

.13 

.14 

2700 

823 

.02 

.05 

.07 

.09 

.11 

.12 

.13 

.14 

.14 

2800 

853 

.03 

.05 

.07 

.09 

.11 

.13 

.14 

.14 

.15 

2900 

884 

.03 

.05 

.08 

.10 

.12 

.13 

.14 

.15 

.15 

3000 

914 

.03 

.05 

.08 

.10 

.12 

.14 

.15 

.15 

.16 

3100 

945 

.03 

.06 

.08 

.10 

.12 

.14 

.15 

.16 

.16 

3200 

975 

.03 

.06 

.08 

.11 

.13 

.14 

.16 

.16 

.17 

3300 

1006 

.03 

.06 

.09 

.11 

.13 

.15 

.16 

.17 

.17 

3400 

1036 

.03 

.06 

.09 

.11 

.14 

.15 

.17 

.17 

.18 

3500 

1067 

.03 

.06 

.09 

.12 

.14 

.16 

.17 

.18 

.18 

3600 

1097 

.03 

.06 

.09 

.12 

.14 

.16 

.18 

.18 

.19 

3700 

1128 

.03 

.07 

.10 

.12 

.15 

.17 

.18 

.19 

.19 

3800 

1158 

.03 

.07 

.10 

.13 

.15 

.17 

.19 

.20 

.20 

3900 

1189 

.04 

.07 

.10 

.13 

.16 

.18 

.19 

.20 

.20 

4000 

1219 

.04 

.07 

.10 

.13 

.16 

.18 

.20 

.21 

.21 

4100 

1250 

.04 

.07 

.11 

.14 

.16 

.19 

.20 

.21 

.21 

4200 

1280 

.04 

.07 

.11 

.14 

.17 

.19 

.21 

.22 

.22 

4300 

1311 

.04 

.08 

.11 

.14 

.17 

.19 

.21 

.22 

.22 

4400 

1341 

.04 

.08 

.11 

.15 

.18 

.20 

.22 

.23 

.23 

4500 

1372 

.04 

.08 

.12 

.15 

.18 

.20 

.22 

.23 

.23 

4600 

1402 

.04 

.08 

.12 

.15 

.18 

.21 

.23 

.24 

.24 

4700 

1433 

.04 

.08 

.12 

.16 

.19 

.21 

.23 

.24 

.24 

4800 

1463 

.04 

.09 

.13 

.16 

.19 

.22 

.24 

.25 

.25 

4900 

1494 

.04 

.09 

.13 

.16 

.20 

22 

.24 

.25 

.26 

5000 

1524 

.05 

.09 

.13 

.17 

.20 

!23 

.24 

.26 

.26 

5100 

1554 

.05 

.09 

.13 

.17 

.20 

.23 

.25 

.26 

.27 

5200 

1585 

.05 

.09 

.14 

.17 

.21 

.23 

.25 

.27 

.27 

5300 

1615 

.05 

.09 

.14 

.18 

.21 

.24 

.26 

.27 

.28 

5400 

1646 

.05 

.10 

.14 

.18 

.22 

.24 

.26 

.28 

.28 

5500 

1676 

.05 

.10 

.14 

.18 

.22 

.25 

.27 

.28 

.29 

132 


TJ.   S.   COAST   AND  GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 

Reduction  of  latitude  to  sea  level — Continued. 


0 
Jl 

5° 
85° 

10" 
80° 

15° 
75° 

20° 
70° 

25° 
65° 

30° 
60° 

35° 
55° 

40°. 
50° 

45° 

Feet 

Meters 

// 

// 

// 

// 

// 

// 

// 

// 

// 

5600 

1707 

0.05 

0.10 

0.15 

0.19 

0.22 

0.25 

0.27 

0.29 

0.29 

5700 

1737 

.05 

.10 

.15 

.19 

.23 

.26 

.28 

.29 

.30 

5800 

1768 

.05 

.10 

.15 

.19 

.23 

.26 

.28 

.30 

.30 

5900 

1798 

.05 

.11 

.15 

.20 

.24 

.27 

.29 

.30 

.31 

6000 

1829 

.05 

.11 

.16 

.20 

.24 

.27 

.29 

.31 

.31 

6100 

1859 

.06 

.11 

.16 

.20 

.24 

.28 

.30 

.31 

.32 

6200 

1890 

.06 

.11 

.16 

.21 

.25 

.28 

.30 

.32 

.32 

6300 

1920 

.06 

.11 

.16 

.21 

.25 

.28 

.31 

.32 

.33 

6400 

1951 

.06 

.11 

.17 

.21 

.26 

.29 

.31 

.33 

.33 

6500 

1981 

.06 

.12 

.17 

.22 

.26 

.29 

.32 

.33 

.34 

6600 

2012 

.06 

.12 

.17 

.22 

.26 

.30 

.32 

.34 

.34 

6700 

2042 

.06 

.12 

.17 

.22 

.27 

.30 

.33 

.34 

.35 

6800 

2073 

.06 

.12 

.18 

.23 

.27 

.31 

.33 

.35 

.35 

6900 

2103 

.06 

.12 

.18 

.23 

.28 

.31 

.34 

.35 

.36 

7000 

2134 

.06 

.12 

.18 

.23 

.28 

.32 

.34 

.36 

.36 

7100 

2164 

.06 

.13 

.19 

.24 

.28 

.32 

.35 

.36 

.37 

7200 

2195 

.07 

.13 

.19 

.24 

.29 

.33 

.35 

.37 

.38 

7300 

2225 

.07 

.13 

.19 

.24 

.29 

.33 

.36 

.37 

.38 

7400 

2256 

.07 

.13 

.19 

.25 

.30 

.33 

.36 

.38 

.39 

7500 

2286 

.07 

.13 

.20 

.25 

.30 

.34 

.37 

.38 

.39 

7600 

2316 

.07 

.14 

.20 

.25 

.30 

.34 

.37 

.39 

.40 

7700 

2347 

.07 

.14 

.20 

.26 

.31 

.35 

.38 

.40 

.40 

7800 

2377 

.07 

.14 

.20 

.26 

.31 

.35 

.38 

.40 

.41 

7900 

2408 

.07 

.14 

.21 

.23 

.32 

.36 

.39 

.41 

.41 

8000 

2438 

.07 

.14 

.21 

.27 

.32 

.36 

.39 

.41 

.42 

8100 

2469 

.07 

.14 

.21 

.27 

.32 

.37 

.40 

.42 

.42 

8200 

2499 

.07 

.15 

.21 

.27 

.33 

.37 

.40 

.42 

.43 

8300 

2530 

.08 

.15 

.22 

.28 

.33 

.37 

.41 

.43 

.43 

8400 

2560 

.08 

.15 

.22 

.28 

.34 

.38 

.41 

.43 

.44 

8500 

2591 

.08 

.15 

22 

.28 

.34 

.38 

.42 

.44 

.44 

8600 

2621 

.08 

.15 

.22 

.29 

.34 

.39 

.42 

.44 

.45 

8700 

2652 

.08 

.16 

.23 

.29 

.35 

.39 

.43 

.45 

.45 

8800 

2682 

.08 

.16 

.23 

.29 

.35 

.40 

.43 

.45 

.46 

8900 

2713 

.08 

.16 

.23 

.30 

.36 

.40 

.44 

.46 

.46 

9000 

2743 

.08 

.16 

.23 

.30 

.36 

.41 

.44 

.46 

.47 

9100 

2774 

.08 

.16 

.24 

.30 

.36 

.41 

.45 

.47 

.47 

9200 

2804 

.08 

.16 

.24 

.31 

.37 

.42 

.45 

.47 

.48 

9300 

2835 

.08 

.17 

.24 

.31 

.37 

.42 

.46 

.48 

.48 

9400 

2865 

.09 

.17 

.24 

.31 

.38 

.42 

.46 

.48 

.49 

9500 

2896 

.09 

.17 

.25 

.32 

.38 

.43 

.47 

.49 

.50 

9600 

2926 

.09 

.17 

.25 

.32 

.38 

.43 

.47 

.49 

.50 

9700 

2957 

.09 

.17 

.25 

.32 

.39 

.44 

.48 

.50 

.51 

9800 

2987 

.09 

.17 

.26 

.33 

.39 

.44 

.48 

.50 

.51 

9900 

3018 

.09 

.18 

.26 

.33 

.40 

.45 

.48 

.51 

.52 

10000 

3048 

.09 

.18 

.26 

.33 

.40 

.45 

.49 

.51 

.52 

CORRECTION  FOR  VARIATION  OF  POLE. 

The  reduction  to  the  mean  position  of  the  pole  is  derived  from  the  provisional  results 
published  by  the  Latitude  Service  of  the  International  Geodetic  Association.  (See  p.  85.) 

DISCUSSION   OF  ERRORS. 

In  discussing  the  errors  of  zenith  telescope  observations  it  is  desirable  to  consider  separately, 
as  on  page  48,  the  external  errors,  observer's  errors  and  instrumental  errors. 

The  principal  external  errors  are  those  arising  from  errors  in  the  adopted  declinations  and 
those  due  to  abnormal  refraction. 


DETERMINATION   OF   LATITUDE.  133 

The  adopted  declinations  used  in  the  computation  necessarily  have  probable  errors  which 
are  sufficiently  large  to  furnish  much,  often  a  half,  of  the  error  of  the  computed  latitude.  This 
arises  from  the  fact  that  a  good  zenith  telescope  gives  results  but  little,  if  any,  inferior  in  accuracy 
to  those  obtained  with  the  large  instruments  of  the  fixed  observatories  which  were  used  in  deter- 
mining the  declinations. 

Of  the  stars  observed  at  thirty-six  latitude  stations,  nearly  on  the  thirty-ninth  parallel, 
between  1880  and  1898,  the  average  value  of  e~  derived  from  the  mean  place  computations 
was  ±o".16  and  the  extreme  values  were  ±0".12  and  ±0".23.  The  average  probable  error 
of  the  declination  of  a  star  in  1900  as  given  for  the  6188  stars  in  the  Boss  catalogue  is  about 
±0".18,  and  hence  the  average  value  of  e  «from  the  Boss  stars  would  be  about  ±0".13.  These 
figures  furnish  a  good  estimate  of  the  accidental  errors  to  be  expected  from  the  adopted  declina- 
tions. To  estimate  the  constant  errors  to  be  expected  from  this  source  is  a  rather  difficult 
matter.  The  principal  constant  error  in  declination  to  be  feared  is  that  arising  from  errors  in 
the  adopted  systematic  corrections  applied  to  the  separate  catalogues  of  observed  places.  The 
three  principal  researches  in  regard  to  these  systematic  corrections  have  been  made  by  Profs. 
Lewis  Boss,  A.  Auwers,  and  Simon  Newcomb.  Judging  by  the  differences  between  the  results 
of  these  three  researches,  the  constant  error  in  the  mean  declinations  based  upon  Professor 
Boss's  researches,  may  possibly  be  as  great  as  0".3,  but  is  probably  much  smaller  than  that. 

In  regard  to  errors  arising  from  abnormal  refraction  it  should  be  noted  that  only  the  dif- 
ference of  refraction  of  the  two  stars  of  a  pair  enters  the  computed  result.  The  errors  in  the 
computed  differential  refractions  are  probably  very  small  when  all  zenith  distances  are  less 
than  45°  and  when  care  is  taken  to  avoid  local  refraction  arising  from  the  temperature  inside 
the  observatory  being  much  above  that  outside,  or  from  masses  of  heated  air  from  chimneys  or 
other  powerful  sources  of  heat  near  the  observatory.  If  there  were  a  sensible  tendency,  as 
has  been  claimed,  for  all  stars  to  be  seen  too  far  north  (or  south)  on  certain  nights,  because  of  the 
existence  of  a  barometric  gradient,  for  example,  it  should  be  detected  by  a  comparison  of  the 
mean  results  on  different  nights  at  the  same  station.  The  conclusion  from  many  such  compar- 
isons made  by  Prof.  John  F.  Hayford  is  that  the  variation  in  the  mean  results  from  zenith 
telescope  measurements  from  night  to  night  is  about  what  should  be  expected  from  the  known 
accidental  errors  of  observation  and  declination;  or,  in  other  words,  that  if  there  are  errors 
peculiar  to  each  night  they  are  exceedingly  small.1 

The  observer's  errors  are  those  made  in  bisecting  the  star  and  in  reading  the  level  and 
micrometer.  Errors  due  to  unnecessary  longitudinal  pressure  on  the  head  of  the  micrometer 
may  also  be  placed  in  this  class. 

Indirect  evidence  indicates  that  the  error  of  bisection  of  the  star  is  one  of  the  largest  errors 
concerned  in  the  measurement.  The  bisections  should  be  made  with  corresponding  care.  The 
probable  error  of  a  bisection  must  be  but  a  fraction  of  the  apparent  width  of  the  micrometer 
line  if  the  observations  are  to  be  ranked  as  first  class.  It  is  possible  to  substitute  three  or  more 
bisections  for  the  one  careful  bisection  recommended  in  the  directions  for  observing  (p.  110), 
but  it  is  not  advisable  to  do  so.  On  account  of  the  comparative  haste  with  which  such  bisections 
must  be  made,  it  is  doubtful  whether  the  mean  of  them  is  much,  if  any,  more  accurate  than  a 
single  careful  and  deliberate  bisection,  while  the  continual  handling  of  the  micrometer  head, 
which  is  necessary  when  several  bisections  are  made,  tends  to  produce  errors. 

With  care  in  estimating  tenths  of  divisions  on  the  micrometer  head  and  on  the  level  grad- 
uation, each  of  these  readings  may  be  made  with  a  probable  error,  of  ±0.1  division.  If  one  turn 
of  the  micrometer  screw  represents  about  60"  and  one  division  of  the  level  about  I",  such 
reading  would  produce  probable  errors  of  ±0".04  and  ±0".05,  respectively,  in  the  latitude 
from  a  single  observation.  These  errors  are  small,  but  not  negligible,  for  the  whole  probable 
error  of  a  single  observation  arising  from  all  sources  is  often  less  than  ±0".30  and  sometimes  less 
than  ±0".20. 

1  See  Report  of  the  Boundary  Commission  upon  the  Survey  and  Re-marking  of  the  Boundary  between  the  United  States  and  Mexico  West  of 
the  Rio  Grande,  1891  to  1896  (Washington,  1898),  pp.  107-109,  for  one  such  comparison. 


134  U.   S.   COAST   AND   GEODETIC    SUKVEY   SPECIAL   PUBLICATION    NO.   1.4. 

While  reading  the  level  the  observer  should  keep  in  mind  that  a  very  slight  unequal  or 
unnecessary  heating  of  the  level  tube  may  cause  errors  several  times  as  large  as  the  mere  reading 
error  indicated  above,  and  that  if  the  bubble  is  found  to  be  moving,  a  reading  taken  after  allow- 
ing it  to  come  to  rest  deliberately  may  not  be  pertinent  to  the  purpose  for  which  it  was  taken. 
The  level  readings  are  intended  to  fix  the  position  of  the  telescope  at  the  instant  when  the  star 
was  bisected. 

It  requires  great  care  in  turning  the  micrometer  head  to  insure  that  so  little  longitudinal 
force  is  applied  to  the  screw  that  the  bisection  of  the  star  is  not  affected  by  it.  Such  a  displace- 
ment of  1-4000  of  an  inch  in  the  position  of  the  micrometer  line  relative  to  the  objective  produces 
an  apparent  change  of  more  than  1"  in  the  position  of  a  star  if  the  focal  length  of  the  telescope 
is  less  than  50  inches.  The  whole  instrument  being  elastic,  the  force  required  to  produce  such 
a  displacement  is  small.  An  experienced  observer  has  found  that  hi  a  series  of  his  latitude 
observations,  during  which  the  level  was  read  both  before  and  after  the  bisections  of  the  star, 
the  former  readings  continually  differed  from  the  latter,  from  0".l  to  0".9,  nearly  always  in 
one  direction.1 

Among  the  instrumental  errors  may  be  mentioned  those  due  (1)  to  an  inclination  of  the 
micrometer  line  to  the  horizon;  (2)  to  error  in  the  adopted  value  of  one  division  of  the  level; 
(3)  to  inclination  of  the  horizontal  axis;  (4)  to  erroneous  placing  of  the  azimuth  stops;  (5)  to 
error  of  collimation;  (6)  to  the  instability  of  the  relative  positions  of  different  parts  of  the 
instrument;  (7)  to  the  irregularity  of  the  micrometer  screw;  (8)  to  the  error  of  the  adopted 
mean  value  of  one  turn  of  the  micrometer  screw. 

The  first  of  these  sources  of  error  must  be  carefully  guarded  against,  as  indicated  on  page  106, 
as  it  tends  to  introduce  a  constant  error  into  the  computed  latitudes.  The  observer,  even  if  lie 
attempts  to  make  the  bisection  in  the  middle  of  the  field  (horizontally),  is  apt  to  make  it  on 
one  side  or  the  other,  according  to  a  fixed  habit.  If  the  line  is  inclined,  his  micrometer  readings 
are  too  great  on  all  north  stars  and  too  small  on  all  south  stare,  or  vice  versa. 

The  error  arising  from  an  erroneous  level  value  is  smaller  the  smaller  are  the  level  correc- 
tions and  the  more  nearly  the  plus  and  minus  corrections  balance  each  other.  If  the  observer 
makes  it  his  rule  whenever  the  record  shows  a  level  correction  of  more  than  one  division  to 
correct  the  inclination  of  the  vertical  axis  between  pairs,  this  error  will  be  negligible.  Little 
time  is  needed  for  this  if  the  observer  avoids  all  reversals  by  simply  manipulating  a  foot-screw 
so  as  to  move  the  bubble  as  much  to  the  northward  (or  the  southward)  as  the  record  indicates 
the  required  correction  to  be. 

The  errors  from  the  third,  fourth,  and  fifth  sources  may  easily  be  kept  within  such  limits 
as  to  be  negligible.  An  inclination  of  1  minute  in  the  horizontal  axis,  or  an  error  of  that  amount 
in  either  collimation  or  azimuth,  produces  only  about  0".  01  error  in  the  latitude.  All  three 
of  these  adjustments  may  easily  be  kept  well  within  this  limit. 

The  errors  arising  from  instability  may  be  small  upon  an  average,  but  they  undoubtedly 
become  large  at  times  and  produce  some  of  the  largest  residuals.  One  of  the  most  important 
functions  of  the  observer  is  to  guard  against  them  by  protecting  the  instrument  from  sudden 
temperature  changes  and  from  shocks  and  careless  or  unnecessary  handling,  and  by  avoiding 
long  waits  between  the  two  stars  of  a  pair.  The  closer  the  agreement  in  temperature  between 
the  observing  room  and  the  outer  air  the  more  secure  is  the  instrument  against  sudden  and 
unequal  changes  of  temperature. 

Most  micrometer  screws  now  used  are  so  regular  that  the  uneliminated  error  in  the  mean 
result  for  a  station  arising  from  the  seventh  source  named  above  is  usually  regligible.  Irregu- 
larities of  sufficient  size  to  produce  a  sensible  error  in  the  mean  result  may  be  readily  detected 
by  inspection  of  the  computation  of  micrometer  value  if  that  computation  is  made  as  indicated 
on  pages  126-128.  The  two  forms  of  irregularity  most  frequently  detected  in  modern  screws  on 
our  latitude  instruments  are  those  with  a  period  of  one  turn  anil  those  of  such  a  form  that  the 
value  of  one  turn  increases  continuously  from  one  end  of  the  screw  to  the  other.  The  periodic 
irregularity  operates  mainly  to  increase  the  computed  probable  error  of  observation  and  must 

1  U.  S.  Coast  and  Geodetic  Survey  Report,  1892,  part  2,  p.  58. 


DETERMINATION   OF    LATITUDE. 


135 


be  quite  large  to  have  any  sensible  effect  upon  the  computed  mean  value  of  the  latitude.  If 
the  value  of  the  screw  increases  continuously  and  uniformly  from  one  end  to  the  other,  the 
computed  results  will  be  free  from  any  error  arising  from  this  source,  provided  all  settings  are 
made  so  that  the  mean  of  the  two  micrometer  readings  upon  a  pair  falls  at  the  middle  of  the 
screw.  If  this  condition  is  fulfilled  within  one  turn  for  each  pair,  the  error  in  the  mean  result 
will  usually  be  negligible.  If  the  settings  are  not  so  made,  it  may  be  necessary  to  compute  and 
apply  a  correction  for  the  irregularity. 

Evidence  has  already  been  presented  on  pages  126-130  to  show  that  it  is  difficult  to  obtain 
the  actual  mean  micrometer  value.  It  is  important,  therefore,  to  guard  against  errors  arising 
from  the  eighth  source  by  selecting  such  pairs  that  the  plus  and  minus  micrometer  differences 
actually  observed  at  a  station  shall  balance  as  nearly  as  possible.  The  final  result  will  be  free 
from  error  from  this  source  if  the  weighted  mean  of  the  micrometer  differences,  the  signs  being 
preserved,  is  zero.  The  only  effect  of  the  error  in  the  mean  micrometer  value  in  that  case  is  to 
slightly  increase  the  computed  probable  errors.  The  weights  are  not,  however,  usually  known 
during  the  progress  of  the  observations.  If  the  indiscriminate  mean  of  the  micrometer  differ- 
ences for  each  pair,  taken  with  respect  to  the  signs,  is  made  less  than  one  turn  at  a  station,  the 
error  of  the  mean  result  from  this  source  will  usually  be  less  than  its  computed  probable  error. 

THE  ECONOMICS  OF  LATITUDE  OBSERVATIONS. 

Two  questions  imperatively  demand  an  answer  under  this  heading.  What  ratio  of  num- 
ber of  observations  to  number  of  pairs  will  give  the  maximum  accuracy  for  a  given  expenditure 
of  money  and  tune  ?  What  degree  of  accuracy  in  the  mean  result  for  the  station  is  it  desirable 
and  justifiable  to  strive  for'? 

The  answer  to  the  first  question  depends  upon  the  relative  magnitude  of  the  accidental 
errors  of  declination  and  of  observation.  At  36  stations  nearly  on  the  thirty-ninth  parallel, 
at  which  latitude  observations  have  been  made  since  the  beginning  of  1880,  the  average  value 
of  e#,  the  probable  error  of  the  mean  of  two  declinations  (derived  from  the  mean  place  com- 
putations), is  ±0".16  and  the  extreme  values  were  ±0".12  and  ±0".23.  At  37  stations 
occupied  with  zenith  telescopes  along  the  thirty-ninth  parallel  the  extreme  values  of  e,  the 
probable  error  of  a  single  observation,  were  ±0".16  and  ±0".98,  and  at  about  one-half  of 
the  stations  it  was  less  than  ±0".42.1  Similarly,  at  43  stations  along  that  parallel  occupied 
with  meridian  telescopes  e  was  less  than  ±0".45  at  one-half  the  stations,  and  the  extreme 
values  were  ±0".21  and  ±1".27.  In  the  light  of  these  figures  one  may  use  the  following  table 
to  determine  the  most  economical  ratio  of  number  of  observations  to  number  of  pairs : 

Weight  to  be  assigned  to  mean  latitude  from  a  single  pair. 


e^  being  assumed  to  be  ±0".16. 


Number  of  observations  on  the  pair 

1 

2 

3 

4 

5 

6 

±0.16 

20 

26 

29 

31.2 

32.6 

33.4 

±0.20 

15 

22 

26 

28.1 

29.8 

31.0 

±0.30 

9 

14 

18 

20.8 

22.9 

24.6 

±0.40 

5.4 

9.5 

12.7 

15.2 

17.4 

19.1 

±0.60 

2.6 

4.9 

6.9 

8.7 

10.2 

11.7 

±0.80 

1.5 

2.9 

4.2 

5.4 

6.5 

7.6 

±1.00 

1.0 

1.9 

2.8 

3.6 

4.4 

5.2 

1  One  thousand  two  hundred  and  seventy-seven  observations  for  variation  of  latitude  at  San  Francisco  in  two  series  gave  e=  ±0".19  and  e=-= 
±0".28.  A  similar  series  at  the  Hawaiian  Islands  in  1891-92,  2434  observations,  gave  e—  ±0".16.  On  the  Mexican  boundary  in  1892-93,  1362 
observations  at  fifteen  stations  gave  e=  ±0".19  to  ±0".38.  All  these  observations  were  made  with  zenith  telescopes.  (See  Coast  and  Geodetic 
Survey  Reports,  1893,  Part  2,  p.  494;  1892  Part  2,  pp.  54  and  158;  1892,  Part  2,  p.  50,  and  Mexican  Boundary  Report,  1891-1896,  p.  101.) 


136  U.   S.   COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.    14. 

The  measure  of  efficiency  of  the  first  observation  is  the  weight  shown  in  the  first  column, 
and  of  each  succeeding  observation  is  the  resulting  increment  of  weight.  Thus,  if  e=  ±0".16, 
the  first  observation  gives  a  weight  of  20,  while  the  second  observation  is  less  than  one-third 
as  efficient,  the  increment  of  weight  being  only  6,  and  the  fifth  and  sixth  observations  com- 
bined are  about  one-ninth  as  efficient  as  the  first  observation.  Stated  otherwise,  the  probable 
error  of  a  single  observation  being  in  this  case  the  same  as  the  probable  error  of  the  mean  of 
two  declinations,  little  is  gained  by  reducing  the  observation  error  while  the  declination  error 
is  allowed  to  remain.  If  e=  ±0".60,  the  table  shows  that  the  second  and  third  observations 
are  each  nearly  as  efficient  as  the  first.  The  larger  is  e  the  less  difference  there  is  between  the 
first  and  succeeding  observations,  but  in  every  case  the  first  observation  is  more  efficient  than 
any  later  observation. 

If  each  observation  after  the  first  involved  the  same  amount  of  time  spent  in  preparation, 
observation,  and  computation  as  the  first,  it  is  evident  that  to  secure  a  maximum  of  accuracy 
for  a  given  expenditure  each  pair  should  be  observed  but  once.  Additional  observations  on 
new  pairs  require  appreciably  more  time  than  the  same  number  of  observations  on  pairs  already 
observed  only  in  the  following  items:  Preparing  the  observing  list,  computing  mean  places, 
and  computing  apparent  places.  Several  observations  per  pair  save  an  appreciable  amount  of 
time  in  the  apparent  place  computation  only  when  the  successive  nights  of  observation  follow 
each  other  so  closely  that  the  apparent  places  on  certain  nights  may  be  obtained  by  interpola- 
tion. (The  interval  over  which  a  straight-line  interpolation  may  be  carried  with  sufficient 
accuracy  is  three  days.) 

After  balancing  this  slight  increase  in  labor  against  the  greater  efficiency  of  the  first  obser- 
vation upon  a  pair  over  any  succeeding  observation,  it  is  believed  that  if  e  is  not  greater  than 
0".40,  each  pah-  should  be  observed  but  once.  If  e  is  much  greater  than  0".40,  two  or  possibly 
even  three  observations  per  pair  may  be  advisable. 

It  is  true  that  if  but  a  single  observation  is  made  upon  each  pair  the  observer  in  the  field 
will  not  be  able  to  determine  his  error  of  observation  accurately  Qie  may  do  so  approximately 
by  assuming  <>„  =  ±0".16),  but  the  field  computation  will  still  perform  its  essential  function 

of  detecting  omissions  and  deficiencies  if  they  exist. 

What  degree  of  accuracy  in  the  mean  result  for  a  station  is  it  desirable  and  justifiable  to 
strive  for?  Omitting  from  consideration  stations  occupied  to  determine  the  variation  of 
latitude,  and  stations  occupied  upon  a  boundary  at  which  one  purpose  of  the  latitude  observa- 
tions is  to  furnish  a  means  of  recovering  the  same  point  again,  the  ordinary  purpose  of  latitude 
observations  in  connection  with  a  geodetic  survey  is  to  determine  the  station  error  in  latitude, 
or,  in  other  words,  to  determine  the  deflection  of  the  vertical,  measured  in  the  plane  of  the 
meridian,  from  the  normal  to  the  spheroid  of  reference  at  the  station.  Broadly  stated,  the 
purpose  of  astronomic  observations  of  latitude  and  longitude  (and  to  a  large  extent  of  azimuth 
also)  in  connection  with  a  geodetic  survey  is  to  determine  the  relation  between  the  actual  figure 
of  the  earth  as  defined  by  the  lines  of  action  of  gravity  and  the  assumed  mean  figure  upon  which 
the  geodetic  computations  are  based.  In  determining  this  relation  three  classes  of  errors  are 
encountered:  The  errors  of  the  geodetic  observations,  the  errors  of  the  astronomic  observa- 
tions, and  the  errors  arising  from  the  fact  that  only  a  few  scattered  astronomic  stations  can 
be  occupied  in  the  large  area  to  be  covered,  and  that  the  station  errors  as  measured  at  these 
few  points  must  be  assumed  to  represent  the  facts  for  the  whole  area.  It  suffices  here  in  regard 
to  errors  of  the  first  class,  which  are  not  within  the  province  of  this  appendix,  to  state  that  they 
are  in  general  of  about  the  same  order  of  magnitude  as  those  of  the  second  class. 

The  average  value  of  the  station  error  in  latitude,  without  regard  to  sign,  at  381  stations 
used  in  the  Supplementary  Investigation  of  the  Figure  of  the  Earth  and  Isostasy,  is  3".8.  An 
examination  of  these  station  errors  shows  that  although  there  is  a  slight  tendency  for  their 
values  for  a  given  region  to  be  of  one  sign  and  magnitude  the  values  at  adjacent  stations  are 
nevertheless  so  nearly  independent  that  the  nonpredictable  rate  of  change  of  the  station  error 
per  mile  is  frequently  more  than  0".l.  Six  stations  within  the  District  of  Columbia  show  an 
irregular  variation  of  station  error  in  latitude  with  a  total  range  of  1".8.  Stating  the  result 


DETERMINATION   OF    LATITUDE.  137 

of  the  examination  in  another  form,  if  the  station  error  at  a  point  is  assumed  to  represent  the 
average  value  of  the  station  error  for  an  area,  and  if  the  error  of  that  assumption  is  to  be  not 
greater  than  ±0".10,  the  area  adjacent  to  the  station  to  which  the  assumption  is  applied  must 
not  be  greater  than  10  square  miles.  If  one  bears  in  mind  that  financial  considerations  so  limit 
the  number  of  latitude  stations  that  in  general  the  above  assumption  must  be  extended  over 
hundreds  of  square  miles,  it  becomes  evident  that  a  probable  error  of  ±0".10  in  the  latitude 
determination  is  all  that  it  is  desirable  or  justifiable  to  strive  for.1  One  observation  upon  each 
of  from  15  to  25  pairs  will  nearly  always  secure  that  degree  of  accuracy,  and  the  observations 
may  be  completed  in  a  single  night. 

As  indicated  in  the  General  Instructions  for  Latitude  Work,  page  104,  paragraphs  3  and  4,  this 
Survey  has  adopted  the  plan  of  using  such  a  number  of  pairs,  observed  but  once,  as  will  make  it 
reasonably  certain  that  the  final  computation  will  give  a  probable  error  not  greater  than  ±0".10 
in  the  resulting  latitude. 

Between  1905  and  1908,  Assistant  W.  H.  Burger  determined  the  latitude  at  63  stations  in 
the  United  States,  making  only  one  observation  on  a  pair  (unless  it  was  found  that  some  mistake 
was  made  on  a  pair,  in  which  case  a  second  observation  was  made  on  it  if  observations  were 
made  on  a  second  night).  The  average  number  of  pairs  observed  per  station  was  16.7,  with  a 
maximum  of  34  pairs  and  a  minimum  of  9  pairs.  The  average  ep  was  ±0".38  and  the  average 
6$  was  ±0".10.  The  average  number  of  nights  on  which  observations  were  made  at  a  station 
was  1.9. 

Assistant  Wm.  Bowie  occupied  7  stations  in  1908.  The  average  number  of  pairs  observed 
per  station  was  15,  with  a  maximum  of  16  and  a  minimum  of  15  pairs.  The,  average  ep  was 
±0".31  and  the  average  e^  was  ±0".08.  Observations  were  made  on  only  8  nights  for  the 
7  stations.  At  only  one  station  were  observations  made  on  more  than  one  night. 

COST  OF  ESTABLISHING  A  LATITUDE  STATION. 

It  is  difficult  to  give  accurately  the  cost  per  station  for  recent  latitude  work  as  usually 
the  parties  were  also  making  observations  for  azimuth.  However,  a  fair  estimate  of  the  cost, 
including  salary  of  the  observer,  for  latitude  stations  by  this  Survey  in  any  except  mountainous 
country  is  about  $200  per  station.  In  a  rough  area  where  pack  animals  would  be  used  exten- 
sively the  cost  might  double  this  estimate.  Where  transportation  is  easy  and  the  stations  not 
distant  from  each  other  the  stations  should  cost  much  less  than  $200  each  if  the  party  remains 
in  the  field  for  long  seasons. 

1  yhe  above  discussion  also  applies,  though  with  less  force,  to  longitude  and  azimuth  observations.    In  both  these  cases  the  errors  of  observation 
are  necessarily  much  larger  than  in  latitude  observations. 


PART    IV. 


DETERMINATION  OF  THE  ASTRONOMIC  AZIMUTH  OF  A  DIRECTION. 


GENERAL  REMARKS. 

Various  methods  are  employed  in  the  Coast  and  Geodetic  Survey  for  determining  astro- 
nomically the  azimuth  of  a  triangulation  line,  or  what  is  the  same  thing,  the  direction  of  that 
line  with  respect  to  the  meridian,  and  there  are,  perhaps,  no  other  geodetic  operations  in  which 
the  choice  of  the  method,  the  perfection  of  the  instrument,  and  the  skill  of  the  observer  enter 
so  directly  into  the  value  of  the  result.  It  is  intended  to  give  here  in  a  concise  form  an  account 
of  several  methods  now  in  use,  and  to  present  the  formulae  as  well  as  specimens  of  record  and 
examples  of  computation.  If  it  is  proposed  to  measure  a  primary  or  subordinate  azimuth,  the 
observer  will  generally  have  the  choice  of  the  method  most  suitable  and  adequate  for  the  pur- 
pose, and  accordingly  provide  himself  with  the  proper  instrument;  yet  frequently  he  may  find 
himself  already  provided  with  an  instrument,  in  which  case  that  method  will  have  to  be  selected 
which  is  compatible  with  the  mechanical  means  at  hand  and  at  the  same  time  insures  the 
greatest  accuracy. 

The  astronomic  azimuth,  or  the  angle  which  the  plane  of  the  meridian  makes  with  the 
vertical  plane  passing  through  the  object  whose  direction  is  to  be  determined,  is  generally 
reckoned  from  the  south  and  in  the  direction  southwest,  etc.  However,  when  circumpolar  stars 
are  observed  it  will  be  found  more  convenient  to  reckon  from  the  north  meridian  and  eastward — - 
that  is,  in  the  same  direction  as  before. 

The  geodetic  azimuth  differs  from  the  astronomic  azimuth.  The  former  is  supposed 
free  from  local  deflections  of  the  plumb  line  or  vertical,  it  being  the  mean  of  several  astronomic 
azimuths,  all  referred  geodetically  to  one  station,  and  it  may  be  supposed  that  in  this  normal 
azimuth  the  several  local  deflections  will  have  neutralized  each  other.  The  astronomic  azimuth 
is,  of  course,  subject  to  any  displacement  of  the  zenith  due  to  local  attraction  or  deflection. 

We  may  distinguish  between  primary  and  secondary  azimuths — the  one  fixing  the  direc- 
tion of  a  side  in  primary  triangulation,  the  other  having  reference  to  sides  of  secondary  or 
tertiary  triangulations  or  to  directions  in  connection  with  the  measure  of  the  magnetic  decli- 
nation. For  the  determination  of  a  primary  azimuth  the  local  time  (sidereal)  must  either  be 
known — as,  for  instance,  when  a  telegraphic  longitude  is  at  the  same  time  determined — or 
special  observations  must  be  made  for  it.  For  subordinate  azimuths,  time  and  azimuth  obser- 
vations may  sometimes  be  made  together,  as  with  the  alt-azimuth  instrument  for  magnetic 
purposes,  in  which  case  the  sun's  limbs  are  usually  observed.  In  refined  work  in  high  latitudes, 
and  for  certain  rare  cases  in  low  latitudes,  the  transit  instrument  is  needed  to  furnish  the  chro- 
nometer correction.  For  primary  azimuths,  in  latitudes  not  greater  than  those  in  the  United 
States,  the  local  time  may  be  found  with  sufficient  accuracy  by  means  of  an  especially  con- 
structed vertical  circle,  used  in  the  Coast  and  Geodetic  Survey,  and  shown  in  illustration  No. 
8.  For  secondary  azimuths,  local  time  may  be  found  by  means  of  sextants  or  alt-azimuth 
instruments. 

PRIMARY  AZIMUTH. 

The  requirements  for  primary  azimuth  are  that  the  astronomic  azimuth  observations  and 

the  necessary  time  observations  should  be  made  using  such  methods,  instruments,  and  number 

of  observations  as  to  make  it  reasonably  certain  that  the  probable  error  of  the  astronomic 

azimuth  does  not  exceed  ±0".50.     It  is  not  desirable  to  spend  much  time  or  money  in  reducing 

138 


No.  18. 


TWELVE-INCH    DIRECTION  THEODOLITE. 


No.  19. 


SEVEN-INCH    REPEATING  THEODOLITE. 


No.  20. 


FOUR-INCH   THEODOLITE. 


DETEBMINATION    OF   AZIMUTH.  139 

the  probable  error  below  this  amount.  At  Laplace  stations  (coincident  triangulation,  longi- 
tude, and  azimuth  stations),  however,  the  astronomic  azimuth  should  be  determined  with  a 
probable  error  not  greater  than  ±0".30  and  the  observations  should  be  made  on  at  least  two 
nights.  When  observations  are  made  to  determine  the  astronomic  azimuth  of  a  line  of  the 
primary  triangulation,  the  azimuth  station  should  coincide  with  a  station  of  the  triangulation 
and  the  mark  used  should  be  some  other  station  of  the  scheme.  In  this  way  the  azimuth  is 
referred  directly  to  one  of  the  lines  of  the  triangulation.  The  probable  error  of  the  azimuth 
of  a  line  obtained  from  an  observed  astronomic  azimuth  on  a  mark  separate  from  the  triangu- 
lation is  greater  than  the  probable  error  of  the  observed  azimuth. 

The  practice  in  the  United  States  Coast  and  Geodetic  Survey  is  for  the  party  on  primary 
triangulation  to  observe  all  necessary  astronomic  azimuths  during  the  progress  of  the  triangu- 
lation. Where  a  direction  instrument  is  used,  the  star  is  often  observed  upon  in  the  regular 
series  of  observations  upon  the  triangulation  stations.  In  such  cases  the  last  object  observed 
upon  in  any  one  series  is  the  star,  and  the  instrument  is  reversed  immediately  after  the  first 
pointing  upon  it.  Where  the  star  is  observed  upon  in  connection  with  two  or  more  triangula- 
tion stations,  the  station  next  preceding  it  is  the  one  to  which  the  astronomic  azimuth  is 

referred. 

INSTRUMENTS. 

So  great  a  variety  of  instruments  is  used  for  azimuth  determinations  that  it  is  of  little 
avail  to  describe  any  particular  instrument  in  detail.  Illustration  No.  18  shows  a  12-inch ' 
direction  theodolite  (No.  146)  made  at  this  office  and  now  in  use  for  the  measurement  of  hori- 
zontal angles  and  azimuths  in  primary  triangulation.  It  carries  a  very  accurate  graduation, 
which  is  read  to  seconds  directly  and  to  tenths  by  estimation  by  three  microscopes.2  A  glass- 
hard,  steel  center  also  contributes  toward  making  this  theodolite  and  others  of  identical  con- 
struction furnish  results  of  a  very  high  degree  of  accuracy.  The  graduation  of  the  horizontal 
circle  on  this  instrument  is  to  5'  spaces.  An  8-inch  repeating  theodolite  reading  to  five  seconds 
by  two  opposite  verniers  is  shown  in  illustration  No.  19.  For  observations  on  the  sun  for  azi- 
muth in  connection  with  magnetic  determinations  a  small  4-inch  theodolite  is  often  used. 
(See  illustration  No.  20.)  This  instrument  reads  to  minutes  on  each  of  two  opposite  verniers. 
The  transit  instruments  and  meridian  telescopes  described  in  connection  with  time  observations 
on  pages  7-8  are  also  frequently  used  for  azimuth  either  in  the  meridian  (p.  160)  or  in  the  vertical 
plane  of  a  circumpolar  star  at  or  near  elongation  (p.  157). 

When  the  azimuth  is  observed  during  the  progress  of  the  primary  triangulation  the  regular 
triangulation  signal  lamps  shown  in  illustrations  Nos.  21  and  22  are  used.  The  smaller  lamp 
can  be  seen  under  average  conditions  to  a  distance  of  about  30  miles.  The  larger  lamp  has  been 
observed  in  the  southwestern  portion  of  the  United  States,  where  the  atmosphere  is  very  clear, 
up  to  distances  of  120  miles.  Where  the  mark  is  only  a  short  distance  from  the  station,  an  ordi- 
nary lantern,  a  bull's  eye  lantern,  or  an  electric  hand  lamp  may  be  used.  In  connection  with  a 
triangulation  along  the  coast  the  lantern  of  a  lighthouse  can  be  used  as  the  mark. 

INSTRUMENT  SUPPORTS. 

While  making  observations  for  a  secondary  azimuth  the  instrument  used  is  xisually  supported 
upon  its  own  tripod,  mounted  upon  stakes  driven  firmly  into  the  ground.  In  primary  triangula- 
tion the  theodolite  is  frequently  mounted  upon  a  tripod  which  may  be  as  much  as  25  or  more 
meters  above  the  ground.  Where  the  instrument  is  not  elevated  it  is  mounted  upon  a  specially 
constructed  wooden  tripod  or  stand  which  has  its  legs  firmly  set  into  the  ground  and  well  braced. 
On  the  top  of  the  legs  is  fitted  a  wooden  cap  usually  2  inches  thick.  On  this  cap  are  fastened 
the  plates  which  receive  the  foot  screws  of  the  theodolite. 

The  structure  shown  in  illustration  No.  23  is  used  to  elevate  the  instrument  in  triangula- 
tion and  azimuth  work.  It  consists  of  a  tripod  on  which  the  instrument  rests  and  a  four-sided 

1  Following  the  usual  practice,  the  size  of  the  theodolite  is  here  designated  by  giving  the  diameter  of  the  graduated  horizontal  circle. 
'  For  a  more  complete  description  of  this  instrument  see  Report  for  1894,  pp.  265-274. 


140  U.   S.   COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   14. 

scaffold  on  which  the  observer  stands.  The  tripod  and  scaffold  do  not  touch  each  other  at  any 
point.  The  top  floor  of  the  scaffold  is  not  needed  on  azimuth  work  and  is  only  used  on  primary 
triangulation  when  there  are  two  observing  parties  working  in  conjunction.  A  complete  descrip- 
tion of  this  type  of  signal  is  given  on  pages  829  to  842  of  Appendix  4,  Report  for  1 903.  Most  of 
the  azimuth  stations  are  in  places  where  it  is  difficult  to  carry  lumber,  and  as  a  result  it  is  usual 
to  have  no  platform  around  the  stand  when  the  instrument  is  only  elevated  above  the  ground 
to  the  height  of  the  observer's  eye.  Where  no  platform  is  used  the  observer  should  be  careful 
not  to  step  close  to  a  leg  of  the  stand  while  making  the  observations  on  the  star.  Such  pre- 
cautions are  not  necessary  to  the  same  extent  while  making  the  observations  on  the  mark 
(or  triangulation  station),  assuming,  of  course,  that  the  mark  is  not  far  from  being  in  the  horizon 
of  the  station.  As  a  result  of  not  using  an  observing  platform  it  may  be  necessary  to  make 
more  observations  to  get  the  desired  degree  of  accuracy  than  if  a  platform  had  been  used.  The 
errors  resulting  from  not  having  a  platform  are  mainly  of  the  accidental  class  and  their  effect 
on  the  final  azimuth  is  small. 

Where  both  azimuth  and  latitude  are  to  be  observed  at  a  station,  but  not  at  the  same  time 
as  the  triangulation  observations,  a  wooden  pier  similar  to  that  shown  in  illustration  No.  24 
has  been  found  satisfactory  in  every  way.  It  was  used  to  a  great  extent  by  former  Assistant 
W.  H.  Burger  and  to  a  limited  extent  by  Assistant  W.  Bowie.  It  will  be  seen  that  the  spread 
and  slope  of  the  legs  of  the  stand  make  it  possible  to  mount  on  it  each  of  the  instruments  in 
turn,  the  top  section  of  the  pier  being  removed  when  used  for  latitude.  The  pier  is  made  as 
if  for  the  azimuth  work,  and  then  the  top  is  sawed  off  at  such  point  as  will  make  the  base  of  the 
pier  of  the  required  height  for  the  latitude  instrument. 

AZIMUTH   MARK. 

When  it  is  necessary  to  elevate  a  signal  lamp  over  a  triangulation  station  used  as  a  mark 
a  number  of  devices  may  be  used.  A  simple  pole  well  guyed  is  frequently  used,  but  this  is  not 
very  satisfactory,  for  it  is  difficult  to  keep  the  support  of  the  lamp  accurately  centered  over  the 
station  mark.  A  device  like  that  shown  in  illustration  No.  25  may  be  used,  and  this  has  the 
advantage  that  the  light  keeper  does  not  have  to  climb  the  pole  when  posting  and  inspecting 
the  lamp.  A  very  satisfactory  and  inexpensive  structure  frequently  used  in  the  United  States 
Coast  and  Geodetic  Survey  is  shown  in  illustration  No.  26.  The  legs,  of  lumber  2  by  4  inches  in 
cross  section,  are  anchored  securely  in  the  ground  and  at  intervals  the  structure  is  guyed  by  wire. 
The  light  keeper  goes  up  the  inside  of  this  signal,  and  near  its  top  there  is  an  opening  leading 
out  to  a  seat.  Such  a  signal  may  be  built  to  a  height  of  140  feet  or  more.  An  acetylene  lamp, 
like  one  of  those  shown  in  illustrations  Nos.  21  and  22,  should  be  posted  at  the  distant  triangula- 
tion station  used  as  the  mark. 

When  the  azimuth  of  a  line  of  the  triangu'ation  is  not  measured  directly,  a  special  azimuth 
mark  is  erected,  which  is  afterwards  referred  to  the  triangulation  by  means  of  horizontal  angles. 
There  has  been  considerable  variety  hi  the  azimuth  marks  so  used,  each  chief  of  party  adapting 
the  mark  to  the  special  conditions  in  which  he  finds  himself  and  to  his  own  convenience.  A 
box  with  open  top  having  in  its  front  face  a  round  hole  or  a  slit  of  suitable  size,  through  which 
the  light  of  a  bull's  eye  or  common  lantern  can  be  shown,  makes  a  satisfactory  mark.  See  illus- 
tration No.  27.  A  white  or  black  stripe  of  paint  or  signal  muslin  can  be  placed  on  the  box,  cen- 
tered over  the  opening,  upon  which  to  make  observations  during  the  day  in  order  to  refer  the 
astronomic  azimuth  of  the  mark  to  a  line  of  the  triangulation. 

The  location  of  the  mark  is  generally  determined,  in  part  at  least,  by  the  configuration  of  the 
ground  surrounding  the  station,  but  it  should  not  be  placed  any  nearer  than  about  one  statute 
mile  in  order  that  the  sidereal  focus  of  the  telescope  may  not  require  changing  between  pointings 
upon  the  star  and  upon  the  mark,  since  any  such  change  is  likely  to  change  the  error  of  collima- 
tion.  Should  the  mark  be  closer  to  the  station  than  one  mile  and  no  change  be  made  in  the 
sidereal  focus  when  pointing  upon  the  mark,  there  would  probably  be  errors  caused  by  parallax. 
If  practicable,  the  mark  should  be  placed  nearly  in  the  horizon  of  the  station  occupied,  in  order 
that  small  errors  of  inclination  of  the  horizontal  axis  of  the  instrument  may  not  affect  the  point- 


a. 

5 


< 
z 
0 


HI 

z 

u 


I- 
u 

u 


DETEEMINATION   OF  AZIMUTH.  141 

ings  upon  the  mark,  and  corresponding  readings  of  the  striding  level  will  be  unnecessary.  In 
choosing  the  position  of  the  mark  it  should  be  kept  in  mind  that  the  higher  the  line  of  sight  to  it, 
above  the  intervening  ground  the  more  steady  the  light  may  be  expected  to  show  and  the  smaller 
the  errors  to  be  expected  from  lateral  refraction. 

SHELTER  FOR  THE   INSTRUMENT. 

An  especially  designed  tent  should  be  used  to  shield  the  instrument  from  the  wind.  Illus- 
trations 16  and  17  show  two  tents  which  have  proved  satisfactory.  The  tent  should  be  only  as 
heavy  as  is  necessary  to  withstand  strong  winds  and  protect  the  instruments  from  rain.  When 
not  in  actual  use  the  instruments  used  for  azimuth  observations  should  be  dismounted  and  placed 
in  their  packing  cases.  Owing  to  the  short  time  during  which  an  azimuth  station  is  occupied 
for  observations  it  is  usually  not  necessary  or  desirable  to  erect  a  wooden  observatory  to  protect 

the  instruments. 

ARTIFICIAL  HORIZON. 

Instead  of  determining  the  inclination  of  the  horizontal  axis  by  readings  of  a  striding  level, 
observations  are  sometimes  taken  upon  the  image  of  the  star  as  seen  reflected  from  the  free 
surface  of  mercury  (an  artificial  horizon)  in  addition  to  the  direct  observations  upon  the  star. 
The  error  in  azimuth  produced  by  the  inclination  of  the  horizontal  axis  is  of  the  same  numerical 
value  for  the  reflected  observations  as  for  the  direct  observations,  but  is  reversed  in  sign,  and 
the  mean  result  is  free  from  error  from  this  source,  provided  the  cross-section  of  each  pivot  is 
circular,  or  at  least  that  the  two  pivots  have  similar  cross-sections  similarly  placed.  Considerable 
care  and  ingenuity  is  necessary  to  protect  the  mercury  effectually  against  tremors  and  against 
wind,  either  of  which  will  by  disturbing  the  mercury  surface  make  the  reflected  star  image  so 
unsteady  as  to  make  accurate  pointing  upon  it  difficult  or  impossible.  A  glass  roof  over  the 
mercury  to  protect  it  from  the  wind  should  never  be  employed  in  connection  with  azimuth 
observations,  since  reversal  of  it  does  not  sufficiently  correct  for  errors  arising  from  refraction  at 
the  glass.  Large  boxes,  or  tubes  of  considerable  size,  with  their  openings  covered  with  mosquito 
netting,  have  proved  the  most  satisfactory  protection  of  the  mercury  against  the  wind. 

It  is  believed  that  the  lateral  refi  action  of  the  direct  and  reflected  ray,  when  the  mercury  is 
set  on  the  ground,  may  introduce  uncertain  and  possibly  large  errors  into  the  azimuth.  This 
trouble  can  be  avoided  by  placing  the  artificial  horizon  on  a  stand  nearly  as  high  as  the  theodolite. 
This,  however,  can  not  be  done  with  the  direction  theodolite  (except  in  very  low  latitudes). 
The  artificial  horizon  can  not  be  used  in  high  latitudes  when  making  observations  on  Polaris,  as 
the  horizontal  circle  of  the  theodolite  would  intercept  the  reflected  ray. 

POINTING   LINES. 

The  pointings  in  azimuth  observations  are  usually  taken  by  using  either  a  single  vertical 
line  in  a  reticle  (or  attached  to  a  micrometer)  or  a  pair  of  parallel  vertical  lines  about  20" 
(of  arc)  apart.  The  first  has  the  advantage  over  the  second  that  it  does  not  involve  the  necessity 
of  bisecting  a  space  by  eye,  as  the  observation  consists  simply  of  noting  when  the  star  image 
appears  symmetrical  with  respect  to  the  line.  On  the  other  hand,  it  has  the  disadvantage  that 
frequently  when  a  very  bright  star  (or  light)  is  observed  the  line  appears  to  be  "burned  off" 
near  the  star  image;  that  is,  it  becomes  invisible  because  of  its  comparative  faintness,  and  the 
pointing  is  correspondingly  uncertain.  So  also  if  a  very  faint  star  (or  light)  is  observed  its 
image  may  nearly  or  completely  disappear  behind  the  line  and  so  make  accurate  pointing 
difficult.  For  many  stars  of  intermediate  degrees  of  brightness  one  or  the  other  of  these  diffi- 
culties exists  to  a  greater  or  less  degree.  If  two  vertical  hnes  are  used  and  the  distance  between 
them  is  properly  chosen  these  two  difficulties  will  be  avoided  and  both  star  (or  mark)  and  lines 
will  always  be  distinctly  visible  at  the  same  instant.  The  observation  now  consists  in  noting 
when  the  image  of  the  star  (or  mark)  bisects  the  space  between  the  two  hnes.  This  process  is 
probably  but  slightly  less  accurate  under  any  conditions  of  brightness  than  the  direct  bisection 


142  U.   S.   COAST  AND  GEODETIC   SURVEY   SPECIAL  PUBLICATION   NO.   H. 

of  a  star  image  under  the  most  favorable  conditions  as  to  brightness.  In  measuring  horizontal 
angles  and  azimuths  in  Colorado,  Utah,  and  Nevada,  along  the  thirth-ninth  parallel,  and  on  all 
primary  triangulation  on  the  ninety-eighth  meridian  since  1901,  and  on  the  Texas-California 
arc  of  primary  triangulation,  two  vertical  lines  about  20"  apart  were  used. 

During  the  progress  of  the  triangulation  along  the  western  part  of  the  thirty-ninth  parallel, 
observations  were  made  at  times  upon  Polaris  in  daylight  to  determine  the  astronomic  azimuth, 
This  is  a  satisfactory  method  and  occasionally  is  convenient  for  the  observer. 

GENERAL  CONSIDERATIONS. 

Let  the  hour  angle  (<),  declination  (d),  and  latitude  (<p)  be  slightly  in  error  by  the  quantities 
dt,  dd,  and  dtp,  and  let  dA  equal  their  effect  upon  the  azimuth  (A)  ;  then,  in  general,  it  will  be 
seen  that,  all  other  circumstances  being  equal,  dA  increases  as  the  zenith  distance  (£)  decreases; 
for  a  star  near  the  pole  and  for  a  latitude  not  too  high  a  small  error  in  time  and  in  latitude  has 
but  a  slight  effect  upon  the  azimuth,  and  in  the  case  of  a  circumpolar  star  at  elongation  (when 
the  parallactic  angle  is  90°)  a  small  error  in  time,  dt,  will  not  affect  the  azimuth;  but  small 
errors  in  declination,  dd,  and  in  latitude,  d<p,  then  attain  nearly  their  maximum  effect  upon  the 
azimuth.  If  observations  are  made  upon  a  circumpolar  star  (8>cp]  at  the  eastern  and  at  the 
western  elongation,  effects  of  dd  and  dg>  will  disappear  in  the  combination  of  the  two  results  ; 
this,  therefore,  is  the  most  favorable  condition  for  observing.  In  general,  effects  of  dd  and  d<p 
disappear  in  mean  results  of  observations  of  equal  and  opposite  azimuths.  In  observations 
on  a  circumpolar  star  in  the  meridian  the  effect  of  a  small  error  in  time  and  in  right  ascension 
may  be  eliminated  by  a  combination  of  results  from  upper  and  lower  culminations;  for  a  star 
in  the  meridian  the  quantities  dd  and  d<p  do  not  enter  in  the  azimuth.  If  the  object  to  be 
observed,  star  (or  sun),  is  of  great  polar  distance  (also  d<  <f>),  and  if  S  is  positive,  the  best  time 
for  observing  is  before  the  eastern  transit,  or  after  the  western  transit  over  the  prime  vertical, 
when  the  change  in  azimuth  with  respect  to  time  is  a  minimum,  but  the  star  (or  sun)  should 
not  be  too  near  the  zenith  nor  be  so  low  as  to  be  affected  by  changes  of  refraction;  if  3  is  negative, 
the  star  (or  sun)  should  be  observed  some  distance  from  the  meridian.1 

These  considerations  have  led  to  the  plan  of  making  first-class  azimuth  observations  almost 
exclusively  upon  the  close  circumpolars  ct,  S,  and  ].  Ursse  Minoris  and  51  Cephei.  The  apparent 
places  of  these  four  stars  are  given  in  the  American  Ephemeris  for  every  day  of  the  year.  Illus- 
tration No.  28  will  assist  in  readily  finding  the  two  fainter  stars  ^  Ursse  Minoris  and  51  Cephei, 
which  barely  become  visible  to  the  naked  eye  under  the  most  favorable  circumstances;  it  also 
shows  that  when  d  Ursse  Minoris  and  51  Cephei  culminate  on  either  side  of  the  pole,  Polaris  is 
not  far  from  its  elongation;  and,  likewise  when  the  pole  star  culminates,  the  other  two  are  on 
opposite  sides  of  the  meridian,  near  their  elongations.  A  similar  approximate  relation  exists 
between  a  and  A  Ursse  Minoris.  Polaris  offers  the  advantage  of  being  observable  in  daytime 
with  portable  instruments;  hence  it  may  be  observed  at  eastern  and  western  elongations,  or 
at  upper  and  lower  culminations,  provided  the  sun  be  not  too  high;  A  Ursse  Minoris,  from  its 
greater  proximity  to  the  pole  and  its  smaller  size,  presents  to  the  larger  instruments  a  finer  and 
steadier  object  for  bisection  than  Polaris;  51  Cephei  is  also  advantageously  used  on  account  of 
its  small  size.  The  star  B.  A.  C.  No.  4165,  shown  on  the  diagram,  was  proposed  and  used  for 
azimuth  work  by  Assistant  G.  Davidson.  The  apparent  processional  motion  of  the  pole  in 
100  years  is  indicated  by  the  direction  and  length  of  the  arrow.  The  sun  is  employed  only  to 
determine  azimuths  of  inferior  accuracy,  generally  in  connection  with  the  determination  of  the 
magnetic  declination. 

'  The  statements  made  in  a  general  and  somewhat  indefinite  form  in  this  paragraph  may  be  stated  in  accurate  mathematical  form  by  deriving 
dA  in  terms  of  it,  dip,d3,  respectively,  from  the  formula 


*n 


cos  <f>  tana—  sin  p  cost 
(see  p.  143),  or  from  the  formulae  used  in  its  derivation. 


No.  25. 


EIGHTY-FOOT  SIGNAL. 


No.  24. 


WOODEN   PIER  USED   FOR  THEODOLITE  AND  ZENITH  TELESCOPE. 


DETEKMINATION   OF   AZIMUTH.  143 

GENERAL  FORMULA. 

Four  methods  of  determining  azimuth  will  be  treated  in  detail  in  this  publication,  namely, 
(1)  the  method  in  which  a  direction  theodolite  is  used,  as  in  the  measurement  of  horizontal 
directions;  (2)  the  method  of  repetitions  with  a  repeating  theodolite;  (3)  the  micrometric 
method,  using  an  eyepiece  micrometer;  (4)  the  determination  of  azimuth  from  time  observa- 
tions with  a  transit  or  meridian  telescope  approximately  in  the  meridian.1  Certain  formulae 
wliich  are  common  to  the  first  three  of  these  methods  will  be  stated  here  for  convenient  reference. 

The  computation  of  the  azimuth  of  a  terrestrial  line  of  sight  from  a  set  of  azimuth  observa- 
tions consists  essentially  of  a  computation  of  the  azimuth  of  the  star  at  the  instant  of  observa- 
tion, a  computation  of  the  horizontal  angle  between  the  star  and  the  mark,  and  the  combination 
of  these  two  results  by  addition  or  subtraction. 

In  the  spherical  triangle  defined  by  the  pole,  the  zenith,  and  a  star,  the  side  zenith-pole  is 
the  co-latitude,  the  side  star-pole  is  the  polar  distance  of  the  star,  and  the  angle  at  the  pole 
is  the  hour  angle  2  or  its  explement.  Starting  from  these  three  as  known  parts,  the  spherical 
triangle  may  be  solved  by  the  ordinary  formulae  of  spherical  trigonometry.  The  solution  to 
obtain  the  azimuth  of  the  star,  which  is  the  angle  of  this  triangle  at  the  zenith,  may,  without 
any  approximations,  be  put  in  the  form 

. sin  t 

cos  <p  tan  d  —  sin  <p  cos  t 

in  which  A  is  the  azimuth  of  the  star  counted  from  the  north  in  a  clockwise  direction,3  and 
the  hour  angle  t  is  counted  westward  from  upper  culmination  continuously  to  24h,  or  360°,  at 
the  next  upper  culmination.  This  is  the  most  convenient  formula  for  use  with  either  of  the 
first  three  methods.  The  first  term  of  the  denominator  changes  very  slowly  and  may  be  tabu- 
lated for  slightly  different  values  of  d  during  the  period  of  observation.  The  second  term,  for 
a  close  circumpolar  star,  may  be  computed  with  sufficient  accuracy  by  five-place  logarithms. 

The  computation  of  the  azimuth  from  this  formula  may  be  considerably  shortened  by 
transforming  it  as  indicated  below  and  using  the  table  given  on  pages  165-173: 4 

sin  t 
tan  A  =  — 


cos  <p  tan  d  —  sin  <p  cos  t 

cot  d  sec  <p  sin  t 
1  —  cot  d  tan  (p  cos  t 

=  —  cot  d  sec  <p  sin  1 1  ^ •  J 

in  which  a  =  cot  d  tan  <p  cos  t. 

The  second  form  of  this  formula  is  about  as  convenient  as  the  first.  It  involves  the  same 
number  of  logarithms  as  the  first  and  one  less  reduction  from  logarithms  to  numbers. 

The  third  form  in  connection  with  the  tables  given  on  pages  165-173  gives  a  much  quicker 
computation  process  than  either  of  the  other  two.  In  using  this  form  and  the  tables,  log  cot 
3  sec  cp  sin  t  must  be  carried  to  six  places  and  log  cot  d  tan  <p  cos  t  to  five  places.  The  most  con- 
venient arrangement  of  the  computation  is  shown  on  page  148.  The  formula  and  tables  involve 
no  approximations,  and  the  only  errors  resulting  from  their  use  are  those  arising  from  the  cast-off 
decimal  places  (logarithms  limited  to  six  places).  These  errors  are  of  the  accidental  class,  and 

i  The  method  of  determining  azimuth  by  observations  upon  the  sun  at  any  hour  angle  is  not  treated  in  this  publication,  because  it  is  used 
mainly  in  making  observations  for  magnetic  declinations  and  a  description  of  it,  with  tables  for  making  the  parallax  and  refraction  corre  rtions,  is 
given  in  "Principal  Facts  of  the  Earth's  Magnetism"  published  in  1909,  and  also  in  "  Directions  for  Magnetic  Measurements"  published  in  1911, 
both  issued  by  the  Coast  and  Geodetic  Survey. 

1  In  this  publication  the  hour  angle  will  be  reckoned  westward  from  zero  at  upper  culmination  (increasing  with  the  lapse  of  time)  to  360°  or  24\ 

» In  astronomic  computations  it  is  more  convenient  to  count  the  azimuth  from  the  north  instead  of  from  the  south,  as  in  geodetic  computa- 
tions. If  the  direction  of  the  count  is  clockwise,  as  here  stated,  to  change  from  one  reckoning  to  the  other  it  is  only  necessary  to  add  or  subtract 
180°. 

«  The  formula  and  the  table  are  both  copied  from  Formiln  und  Hulfsta/dn  fiir  Geographische  Ortabestimmunyen  von  Prof.  Dr.  Th.  Albrecht, 
Leipzig,  1894.  The  range  of  the  table  has,  however,  been  considerably  extended. 


144  U.   S.   COAST  AND  GEODETIC   SUBVEY  SPECIAL   PUBLICATION   NO.   14. 

will  seldom  exceed  0''.04  for  any  case  covered  by  the  table,  and  for  most  observations  made 
below  latitude  50°  the  error  will  not  exceed  0".01.  These  quantities  are  so  small  in  comparison 
with  the  errors  of  observation  as  to  be  negligible.  A  few  observations  made  in  Alaska  may  be 
beyond  the  range  of  the  tables  on  pages  165-173,  and  when  that  is  found  to  be  the  case,  one 
may  easily  substitute  the  second  formula  on  page  .143  for  the  third.1 

To  compute  the  azimuth  of  a  star  at  the  time  of  each  pointing  made  upon  it  during  a  set 
of  observations  is  an  unnecessarily  laborious  process.  If  for  the  hour  angle,  t,  of  the  azimuth 
formula  is  taken  the  mean  of  the  hour  angles  of  the  set,  the  computed  azimuth  is  that  corre- 
sponding to  the  mean  hour  angle,  but  is  not  the  required  mean  of  the  azimuths  corresponding  to  the 
separate  hour  angles,  since  the  rate  of  change  of  the  azimuth  is  continually  varying  because  of 
the  curvature  of  the  apparent  path  of  the  star.  The  difference  between  the  two  quantities  indi- 
cated by  the  italics  is  small,  though  not  usually  negligible,  for  the  interval  of  time  covered  by  a 
set  of  observations.  The  most  convenient  way  of  making  the  computation  for  a  set  of  observa- 
tions is  to  use  the  mean  hour  angle  in  the  azimuth  formula  and  apply  to  the  result  a 

.  1  _2  sin2  i  T 
Curvature  Correction  =  tan  A-Z 


n       sin  1" 

in  which  n  is  the  number  of  pointings  upon  the  star  in  the  set  and  r  for  each  observation  is  the 
difference  2  between  the  time  of  that  observation  and  the  mean  of  the  times  for  the  set.  The 
sign  of  this  curvature  correction  is  always  such  as  to  decrease  numerically  the  azimuth  reckoned 
from  the  north,  or  in  other  words,  if  azimuths  are  counted  clockwise  its  algebraic  sign  will  be  + 
when  the  star  is  west  of  north  and  —  when  the  star  is  east  of  north.  If  the  star  crosses  the 
meridian  during  the  progress  of  a  set  the  curvature  correction  will  ordinarily  be  zero.  The 
formula  is  approximate,  but  for  circumpolars  and  for  the  interval  of  time  usually  covered  by 

2  sin2  \  T 

a  set  of  observations  its  errors  are  negligible.     The  value  of  the  term—  — -TTJ — may  be  found 

sm  i 

on  pages  151-152  of  this  publication.3 

If  the  star  observed  is  Polaris,  a  convenient  rough  check  on  the  computation  may  be 
obtained  from  Table  V  of  the  American  Ephemeris  and  Nautical  Almanac,  entitled  Azimuth  of 
Polaris  at  all  Hour  Angles. 

Because  of  the  rapid  motion  of  the  observer,  due  to  the  rotation  of  the  earth  on  its  axis, 
a  star  is  seen  slightly  displaced  from  its  real  position.  The  required 

Correction  for  Diurnal  Aberration  =  0". 32  °°S  A  COS  * 


cos  h 

The  sign  of  the  correction  is  always  positive  when  applied  to  azimuths  counted  clockwise. 
The  greatest  variation  of  the  correction  from  its  mean  value,  0".32,  for  the  four  circumpolars 
ordinarily  observed  and  for  latitudes  not  greater  than  50°,  is  0".02.  The  correction  for  diurnal 
aberration  need  not  be  applied  to  the  separate  sets  but  simply  to  the  mean  result  for  a  station. 

If  the  horizontal  axis  is  inclined  when  the  pointings  are  made  upon  either  the  star  or  the 
mark  the  corrections  indicated  below  must  be  applied. 

Level  Correction  =  - \(w  +  w')  —  (e  +  e')  tan  h 

if  the  striding  level  carries  a  graduation  numbered  in  both  directions  from  the  middle,     d  is 
the  value  of  one  division  of  the  level  and  w,  e  and  w',  e'  are  the  west  and  east  readings  of  the 

1  Various  other  formulas  for  computing  the  azimuth  of  circumpolar  stars  have  been  proposed  and  used.  Each  of  them  requires  either  the  same 
or  a  greater  time  for  the  computation  than  that  here  given,  when  the  whole  computation,  including  the  preparation  of  the  auxiliary  tables  required 
with  some  of  them,  is  taken  into  account.  As  uniformity  of  practice  is  conducive  to  rapid  computation,  it  is  considered  desirable  that  all  should 
use  the  formula;  given,  and  therefore  no  others  are  here  stated.  It  should  be  noted  that  the  formula  given  is  accurate  and  general;  that  is,  it 
applies  to  any  of  the  close  circumpolars  at  any  hour  angle. 

>  If  a  mean  time  chronometer  is  used,  the  value  I  — ^  1,,T  should  be  increased  by  its  one  hundred  and  eightieth  part. 

«  This  table  was  copied  from  pages  634-637  ot  Doolittle's  Practical  Astronomy.    These  tabular  values  may  be  found  in  various  other  places. 


No.  25. 


STRUCTURE  FOR  ELEVATING  SIGNAL  LAMP  OVER 
TRIANGULATION   STATION   USED  AS   MARK. 


No.  26. 


STRUCTURE  FOR  ELEVATING  SIGNAL  LAMP  OVER  TRIANGULATION 
STATION    USED   AS    MARK. 


No.  27. 


AZIMUTH    MARK. 


DETERMINATION    OF   AZIMUTH.  145 

level  before  and  after  reversing  it.     h  is  the  altitude  of  the  star.     It  is  only  necessary  to  know 
h  approximately — an  occasional  reading  of  the  setting  circle  will  give  it  with  abundant  accuracy, 
If  the  graduation  on  the  striding  level  is  numbered  continuously  in  one  direction  the 


Level  Correction  =  j \(w  —  w')  +  (e  —  er)  tan  h 


in  which  the  primed  letters  refer  to  readings  taken  in  the  position  in  which  the  numbering 
increases  toward  the  east.1 

If  the  mark  is  not  in  the  horizon  of  the  instrument  a  similar  correction,  if  appreciable, 
must  be  applied  to  readings  upon  the  mark,  Ti  now  being  the  altitude  of  the  mark.  Ordinarily 
the  mark  is  so  nearly  in  the  horizon  of  the  instrument  that  tan  Ti  is  nearly  zero  and  the  correc- 
tions required  to  pointings  upon  the  mark  are  negligible. 

The  formula  as  written  gives  the  sign  of  the  correction  to  be  applied  to  the  readings  of  a 
horizontal  circle  of  which  the  numbering  increases  in  a  clockwise  direction.  This  is  also  the 
sign  of  the  correction  to  the  computed  azimuth  (counted  clockwise)  for  level  readings  in  connec- 
tion with  pointings  upon  the  mark,  but  in  connection  with  pointings  upon  the  star  the  sign 
must  be  reversed  to  give  corrections  to  the  computed  azimuth  of  the  mark. 

DIRECTION  METHOD— ADJUSTMENTS. 

The  measurement  of  an  azimuth  by  this  method  is  essentially  similar  to  the  process  of 
measuring  a  difference  of  two  horizontal  directions  with  a  direction  theodolite.  The  quantity 
measured  in  this  case  is  the  difference  of  azimuth  of  a  circumpolar  star  and  a  mark  instead  of 
a  difference  of  azimuth  of  two  triangulation  signals.  The  fact  that  the  azimuth  of  the  star  is 
continually  changing  adds  new  features  to  the  computation,  and  makes  it  necessary  to  know 
the  time  of  each  pointing  upon  the  star.  The  fact  that  the  star  is  at  a  considerable  altitude 
makes  readings  of  the  striding  level  a  necessity  and  decreases  the  accuracy  of  the  measurement 
because  errors  of  inclination  of  the  horizontal  axis  have  a  marked  influence  as  contrasted  with 
their  comparatively  unimportant  effects  upon  the  measurements  of  horizontal  angles  in  a 
triangulation. 

The  adjustments  required  are  identical  with  those  which  are  necessary  when  the  instrument 
is  to  be  used  for  the  measurement  of  horizontal  directions.  The  adjustments  of  the  focus  of 
the  telescope,  of  the  line  of  collimation,  for  bringing  the  vertical  lines  of  the  reticle  into  vertical 
planes,  of  the  setting  circle  (if  used),  and  of  the  strding  level  may  be  made  as  described  in 
connection  with  a  transit  on  pages  14-16.  The  vertical  axis  of  the  instrument  must  be  made 
to  point  as  nearly  as  is  feasible  to  the  zenith  by  bringing  the  striding  level  to  the  proper  reading 
in  each  of  two  positions  at  right  angles  to  each  other. 

The  microscopes  with  which  the  horizontal  circle  is  read  must  be  kept  in  adjustment. 
Ordinarily  it  will  only  be  found  necessary  to  adjust  the  eyepiece  by  pushing  it  hi  or  pulling 
it  out  until  the  most  distinct  vision  is  obtained  of  the  micrometer  lines  and  of  the  circle 
graduation.  If  the  micrometer  lines  are  not  apparently  parallel  to  the  graduation  upon  which 
the  pointing  is  to  be  made,  they  should  be  made  so  by  rotating  the  micrometer  box  about  the 
axis  of  figure  of  the  microscope.  If  to  do  this  it  is  necessary  to  loosen  the  microscope  in 
its  supporting  clamp,  great  caution  is  necessary  to  insure  that  the  distance  of  the  objective 
from  the  circle  of  graduation  is  not  changed.  The  error  of  run  of  the  reading  micrometers 
should  be  kept  small.  In  other  words,  the  value  of  one  turn  of  the  micrometer  in  terms  of 
the  circle  graduation  should  not  be  allowed  to  differ  much  from  its  nominal  value.  The  value 
of  the  micrometer  may  be  adjusted  by  changing  the  distance  of  the  objective  from  the  gradua- 
tion. The  nearer  the  objective  is  to  the  graduation  the  smaller  is  the  value  of  one  turn.  A 
change  in  this  distance  also  necessitates  a  change  in  the  distance  from  the  objective  to  the 
micrometer  lines,  these  lines  and  the  graduation  being  necessarily  at  conjugate  foci  of  the 

'  See  footnote  on  p.  23. 
8136°— 13 10 


146  TT.   S.    COAST   AND  GEODETIC   SUKVEY   SPECIAL   PUBLICATION    NO.    14. 

objective.  This  adjustment  of  the  micrometer  value  is  a  difficult  one  to  make,  but  when  once 
well  made  it  usually  remains  sufficiently  good  for  a  long  period. 

As  stated  on  page  139,  primary  azimuths  are  nearly  always  observed  during  the  progress  of 
the  primary  triangulation,  and  the  same  instrument  is  used  to  make  the  observations  on  the 
azimuth  star  that  is  used  to  determine  the  horizontal  directions  of  the  lines  of  the  triangulation. 
For  a  number  of  years  past  only  the  12-inch  (30  cm.)  direction  theodolites  (described  in  Appen- 
dix 8,  Coast  and  Geodetic  Survey  Report  for  1894)  have  been  used  on  primary  triangulation. 
(See  illustration  No.  18.)  Practically  all  the  observations  for  primary  azimuth  are  made  on 
Polaris.  In  recent  years  the  azimuth  observations  have  been  made  at  the  same  time  that 
horizontal  observations  are  being  made — that  is,  Polaris  is  observed  at  a  setting  of  the  instru- 
ment in  connection  with  one  or  more  of  the  triangulation  stations.  The  observations  on  Polaris 
are  made  at  the  end  of  the  position  in  order  that  the  direct  and  reversed  observations  on  the 
star  may  come  close  together.  Instead  of  determining  the  astronomic  azimuth  of  the  line  used 
as  the  initial  direction  for  the  horizontal  angle  work  it  is  considered  that  the  azimuth  has  been 
determined  of  the  line  observed  over  just  previous  to  the  observations  on  Polaris.  If  at  any 
station  it  is  necessary  to  make  the  observations  for  azimuth  in  connection  with  two  lines  of  the 
triangulation,  then  the  probable  error  of  the  angle  between  the  two  lines  must  be  taken  into 
account  in  deriving  the  probable  error  of  the  azimuth.  When  a  quadrilateral  system  is  used  in 
the  triangulation  and  both  diagonal  lines  are  observed,  then  at  each  station  there  will  be  five 
primary  directions  to  observe. 

Illustration  No.  29  shows  the  lines  radiating  from  such  a  station.  The  station  A,  the  first 
to  the  east  of  Polaris,  is  chosen  as  the  initial  and  the  other  stations  are  observed  in  turn  from 
left  to  right,  and  after  observations  have  been  made  on  E  they  are  made  on  Polaris.  If,  for 
any  reason,  the  line  to  E  is  not  observed  with  the  other  stations  during  observations  for  any- 
one position,  then  Polaris  also  should  not  be  observed.  Later  on  the  instrument  should  be  set 
for  the  missing  position,  and  Polaris  should  be  observed  in  connection  with  station  E. 

The  observer  is  instructed  to  secure  an  accuracy  represented  by  a  probable  error  of  ±0".50 
for  the  greater  portion  of  the  primary  azimuths,  and  the  observations  may  all  be  made  during 
one  night.  This  accuracy  can  usually  be  secured  by  observing  one  set  in  each  of  from  12  to 
16  positions  of  the  instrument.  In  no  case  must  an  azimuth  depend  upon  less  than  10  positions. 

At  some  of  the  triangulation  stations  where  the  accumulated  twist  of  the  triangulation  is 
to  be  determined  by  a  coincident  longitude  and'  azimuth  station  the  azimuth  is  determined 
with  an  accuracy  represented  by  a  probable  error  of  ±0".30,  and  the  observations  are  made 
on  at  least  two  nights. 

DIRECTION  METHOD— EXAMPLE  OF  RECORD  AND  COMPUTATION. 

There  are  shown  below  samples  of  records  of  azimuth  observations  on  Polaris  and  the 
computations.  The  observations  were  carried  on  at  the  same  time  that  observations  of  hori- 
zontal directions  were  made  at  the  primary  triangulation  station,  Sears,  in  Texas.  The  chro- 
nometer correction  and  rate  were  determined  from  observations  with  a  vertical  circle  on  stars 
approximately  on  the  prime  vertical.  Examples  of  the  time  observations  and  computations 
made  at  Sears  for  use  in  the  azimuth  observations  are  shown  on  pages  54  and  55  of  this 
publication. 


No.  28. 


£  URS.MIN. 


XII 


CIRCUMPOLAR  STARS. 


No.  29. 


Polaris 


Static 


DIAGRAM  SHOWING   DIRECTIONS  TO  TRI ANGU  LATION   STATIONS  AND   POLARIS 


DETERMINATION-    OF   AZIMUTH. 


147 


Form  251 


Horizontal  directions. 

[Station,  Sears,  Tex.  (Triangulation  Station).    Observer,  W.  Bowie.    Instrument,  Theodolite  168.    Date,  Doc.  22,  1908.] 


Posi- 
tion 

Objects  observed 

Time 

Tel. 
D  or  R 

Mic. 

Backward 

For- 
ward 

Mean 

Mean 
D 
and  R 

Direc- 
tion 

Remarks 

ft     TO 

0 

, 

,, 

„ 

„ 

1 

Morrison 

8    19 

D 

A 
B 

0 

0 

35 
41 

35 
41 

1   division   of   the 
striding     level  = 

C 

36 

34 

37.0 

4".194 

R 

A 

180 

00 

36 

35 

B 

32 

31 

0 

35 

34 

33.8 

35.4 

00.0 

Buzzard 

D 

A 

53 

30 

43 

42 

B 

41 

42 

C 

34 

33 

39.2 

R 

A 

233 

30 

39 

37 

B 

34 

32 

C 

38 

3S 

36.3 

37.  S 

02.4 

Allen 

D 

A 

no 

14 

61 

62 

B 

57 

55 

C 

61 

59 

59.2 

R 

A 

350 

14 

50 

49 

B 

63 

60 

0 

53 

53 

54.7 

57.0 

21.6 

Polaris 

D 

A 

252 

01 

54 

53 

W                    E 

km      s 

B 

54 

53 

9.3                  28.0 

1    48    35.5 

C 

51 

51 

52.7 

27.  7                    9.  1 

1    51    06.0 



18.4    —  0.5    18.9 

1    49    50.8 

R 

A 

72 

01 

09 

09 

24.9                    6.3 

B 

02 

01 

13.0                  31.7 

C 

10 

08 

06.5 

29.0 



11.9    -13.5    25.4 

-  7.0 

148 


U.   S.   COAST   AND   GEODETIC    SURVEY    SPECIAL   PUBLICATION    NO.   14. 


Form  380. 


Computation  of  azimuth,  direction  method. 

[Station,  Sears,  Tex.    Chronometer,  sidereal  1769.    ^=32°  33  31".    Instrument,  theodolite  168.    Observer,  W.  Bowie.) 


Date,  1908,  position 
Chronometer  reading 
Chronometer  correction 
Sidereal  time 
a  of  Polaris 
t  of  Polaris  (time) 
t  of  Polaris  (arc) 
S  of  Polaris 

Dec.  22,       1 
1    49      50.  8 
4     37.5 
1    45      13.  3 
1    26     41.  9 
0    18     31.4 
4°  37'  51".  0 
88    49      27.  4 

2 
2    01      33.  0 
4     37.  5 
1    56     55.  5 
1    26     41.  9 
0    30      13.  6 
7°  33'  24".  0 

3 
2     16     31.0 
4      37.4 
2     11      53.  6 
1     26      41.  8 
0    45      11.  8 
11°  17'57".0 

4 
2    43      28.  8 
4     37.3 
2    38     51.  5 
1    26     41.  8 
1     12     09.  7 
18°  02'  25".  5 

log  cot  8 
log  tan  <j> 
log  cos  t 

8.  31224 
9.  80517 
9.  99858 

8.  31224 
9.80517 
9.  99621 

8.  31224 
9.  80517 
9.  99150 

8.  31224 
9.  80517 
9.97811 

log  o  (to  five  places) 

8.  11599 

8.  11362 

8.  10891 

8.  09552 

log  cot  8 
log  sec  0 
log  sin  t 

log  ;  
6  1  —  a 

8.  312243 
0.  074254 
8.  907064 

0.  005710 

8.  312243 
0.  074254 
9.  118948 

0.  005679 

8.  312243 
0.  074254 
9.  292105 

0.  005618 

8.  312243 
0.  074254 
9.  490924 

0.  005445 

log  (—tan  A)  (to  6  places) 
A=  Azimuth  of  Polaris,  from  north* 
Difference  in  time  between  D. 
and  R. 
Curvature  correction 

7.  299271 
0    06    50.  8 
m      s 
2    30 
0 

7.  511124 
0    11    09.2 
m      s 
2    00 
0 

7.  684220 
0     16    36.  9 
m     s 
3     18 
0 

7.  882866 
0    26     15.0 
m     s 
1     38 
0 

Altitude  of  Polaris=ft 
1  tan  A=level  factor 

0              /               // 

33     46 
0.701 

0              /               // 

33    46 
0.701 

O              /               // 

33    46 
0.701 

O             /                // 

33     46 
0.701 

Inclination  f 
Level  correction 
Circle  reads  on  Polaris 

-7.0 
-4.9 
252    01    29.  6 

-7.2 
-5.0 
86    58     11.  2 

-7.0 
-4.9 
281    54    27.  0 

-1.8 
-1.3 
116    45    48.  6 

Corrected  reading  on  Polaris 
Circle  reads  on  mark 

252    01    24.  7 
170     14     57.  0 

86    58    06.  2       281     54     22.  1 
5     15    58.2       200    17    42.4 

116     45     47.3 
35     18     45.  4 

Difference,  mark  —  Polaris 
Corrected  azimuth  of  Polaris,  from 
north  * 

278     13    32.  3 

0    06    50.  8 
180    00    00.0 

278     17    52.  0 

0    11    09.  2 
180    00    00.  0 

278     23     20.  3 

0    16    36.  9 
180    00    00.  0 

278     32     58.  1 

0     26     15.  0 
180    00    00.  0 

Azimuth  of  Allen 
(Clockwise  from  south) 

98    06    41.  5 

98     06    42.8         98     06     43.4 

98     06     43.  1 

To  the  mean  result  from  the  above  computation  must  be  applied  corrections  for  diurnal  aberration  and  eccentricity  (if  any)  of  Mark. 

Carry  times  and  angles  to  tenths  of  seconds  only. 

*  Minus,  if  west  of  north. 

t  The  values  shown  in  thjs  line  are  actually  four  times  the  inclination  of  the  horizontal  axis  in  terms  of  level  divisions. 


DETERMINATION   OF   AZIMUTH. 

Summary  of  azimuth  results. 

[Sears,  Tex.,  Dec.  22, 1908.] 


149 


Posi- 
tion 

Azimuth  of  Allen 

V 

i>' 

o         /          // 

1 

98     06    41.  5 

+0.8 

.64 

2 

42.8 

-0.5 

.25 

3 

43.4 

-1.1 

1.21 

4 

43.1 

-0.8 

.64 

5 

39.7 

+2.6 

6.76 

6 

42.7 

-0.4 

.16 

7 

41.6 

+0.7 

.49 

8 

43.3 

-1.0 

1.00 

9 

40.0 

+2.3 

5.29 

10 

45.0 

-2.7 

7.29 

11 

43.3 

-1.0 

1.00 

12 

40.7 

+1.6 

2.56 

X\ 

2=27.  29 

e=  ±0.6745 


/    Iv2 
\  n(n  — 


1) 


±0".31 


The  mean  observed  azimuth 


98°  06'  42".26±0".31. 


Diurnal  aberration  +0.32. 

Correction  for  eccentric  light  +0.04. 

Correction  for  elevation  of  mark  —  0.01 . 

Keduction  to  mean  position  of  pole  *  —  0.29. 

Azimuth  of  the  line  from  Sears  to  Allen  2  =  98  06      42.32  ±0.31. 

DIRECTION  METHOD— EXPLANATION  OF  RECORD  AND  COMPUTATION. 

The  triangulation  stations  and  Polaris  which  were  observed  at  one  setting  of  the  instru- 
ment (in  this  case  position  No.  1)  are  placed  in  the  record  in  the  order  of  their  azimuths  (left 
to  right)  from  the  initial  station,  "Morrison."  The  telescope  in  its  direct  position  is  pointed 
upon  each  station  in  turn  and  finally  upon  Polaris.  The  telescope  is  then  reversed,  and  the 
first  pointing  after  reversal  is  upon  Polaris;  then  pointings  are  made  upon  the  triangulation 
stations  in  the  reverse  order  of  azimuth  (from  right  to  left).  The  readings  in  the  reversed 
position  of  the  telescope  are  placed  directly  under  the  direct  reading.  The  mean  of  the  readings 
in  the  direct  and  in  the  reversed  positions  of  the  telescope  is  used  in  computing  the  direction 
of  a  line  with  reference  to  the  initial  line.  There  are  three  microscope  micrometers  on  the 
instrument  used  in  making  the  observations  at  Sears,  and  at  each  pointing  a  backward  and 
forward  reading  of  each  micrometer  was  made  on  the  two  graduations  of  the  circle  nearest  the 
center  of  the  comb. 

The  mean  run  of  the  micrometers  was  kept  very  small  and  as  the  micrometer  was  placed 
upon  a  different  portion  of  the  five-minute  space  between  successive  graduations,  the  resultant 
effect  of  the  micrometer  run  was  negligible.  The  initial  positions  (minutes  and  seconds)  of  the 
micrometer  wire  on  the  circle  for  the  first  four  positions  were  00'  40",  01'  50",  03'  10",  and 
04'  20".  In  general,  12  or  16  positions  of  the  circle  are  used  for  the  initial  settings  and 
these  readings  of  the  minutes  and  seconds  on  the  initial  are  repeated  in  each  group  of  four 
positions;  that  is,  in  positions  5  to  8,  9  to  12,  and  13  to  16.  It  can  be  shown  that  on  any  object 
the  error  due  to  run  is  practically  zero  in  each  set  of  four  positions  of  the  circle,  if  the  mean 
run  of  the  three  micrometers  with  regard  to  sign  is  less  than  1".0  and  the  run  of  no  one  micrometer 
is  larger  than  3".0.  Special  observations  are  made  in  primary  triangulation  to  determine 
whether  the  run  of  the  micrometers  is  within  these  limits. 


'  See  Astronomische  Nachrichten  No.  4414. 
'Sears  and  Allen  are  triangulation  stations. 


150 


II.   S.   COAST  AND  GEODETIC   SURVEY   SPECIAL  PUBLICATION   NO.   14. 


The  chronometer  time  of  the  observations  on  Polaris  and  also  the  level  readings  are  shown 
in  the  record.  The  time  of  making  an  observation  may  be  noted  by  the  observer  who  picks  up 
and  carries  the  beat  of  the  chronometer,  or  an  assistant  may  note  the  clock  time  upon  a  signal 
from  the  observer.  When  the  latter  method  is  used  the  observer  calls  "Mark"  when  the  star 
is  bisected. 

The  chronometer  corrections  shown  in  the  computations  resulted  from  a  special  series  of 
time  observations  with  the  vertical  circle  at  the  station  (see  pp.  54  and  55). 

The  formula  used  in  making  the  computation  is  the  third  form  of  the  azimuth  formula 

shown  on  page  143.     The  tables  on  pages  165  to  173  which  give  the  logarithm  of  ^  --  were  used  in 

i     a 

the  computations.  Much  time  is  saved  in  such  computations  as  the  above  by  carrying  along  all 
the  different  sets  at  one  time  and  thus  working  along  the  horizontal  lines  of  the  form  shown 
instead  of  down  each  column.  Also  tan  <f>  and  sec  (f>  are  constants  for  the  station,  cos  t  and  sin  t 
may  be  taken  out  at  one  opening  of  the  logarithm  table,  etc.  A  comparison  of  corresponding 
parts  of  different  columns  furnishes  rough  checks  which  serve  to  locate  any  large  errors  quickly. 
The  value  of  one  division  of  the  striding  level  is  4".  194.  In  general,  one  set  like  the  above, 
in  each  of  12  to  16  positions  of  one  of  the  12-inch  theodolites,  will  give  a  probable  error  of 
the  result  less  than  ±0".50.  Even  where  the  observations  for  azimuth  are  made  coincidently 
with  those  for  horizontal  directions  in  a  triangulation  there  is  no  difficulty  in  completing  the 
azimuth  observations  at  a  station  in  one  evening.  For  special  stations  a  probable  error  of  the 
result  of  ±0".30  or  less  must  be  gotten  and  observations  must  be  made  on  more  than  one  night. 
The  general  practice  now  in  the  Coast  and  Geodetic  Survey  is  to  make  only  one  pointing  on  the 
star  in  each  of  the  positions  of  the  telescope  and  therefore  the  correction  for  curvature  of  the 
path  of  the  star  between  the  two  pointings  is  usually  negligible.  When  there  is  a  delay  in 
making  the  second  pointing  the  curvature  correction  should  be  computed  by  the  formula  shown 
on  page  144. 

2  sin2  -ir 

..„    are  given  on  pages  151-152.     The  small  table  shown  below  gives 

Sill    I 


Tabular  values  of 


the  values  of  the  curvature  correction  direct  for  values  of  the  interval,  2r,  between  the  two 
pointings  on  the  star,  from  2  to  7  minutes,  and  azimuths  of  Polaris  less  than  2°  30',  for  use  with 
the  direction  method,  when  only  two  observations  are  made  on  Polaris  for  one  setting  of  the 
instrument. 

Curvature  correction. 


N^   2t 

Azi-    N. 
muthof\ 
Polaris.     \ 

2m 

3» 

*. 

5» 

6m 

7" 

o      / 

// 

ff 

// 

// 

// 

// 

0    10 

.0 

.0 

.0 

.0 

.1 

.1 

0    20 

.0 

.0 

.0 

.1 

.1 

.1 

0    30 

.0 

.0 

.1 

.1 

.2 

.2 

0    40 

.0 

.1 

.1 

.1 

.2 

.3 

0    50 

.0 

.1 

.1 

.2 

.3 

.3 

1    00 

.0 

.1 

.1 

.2 

.3 

.4 

1    10 

.0 

.1 

.2 

.2 

.4 

.5 

1    20 

.0 

.1 

.2 

.3 

.4 

.6 

1    30 

.0 

.1 

.2 

.3 

.5 

.6 

1    40 

.1 

.1 

.2 

.4 

.5 

.7 

1    50 

.1 

.1 

.3 

.4 

.6 

.8 

2    00 

.1 

.2 

.3 

.4 

.6 

.8 

2    10 

.1 

.2 

.3 

.5 

.7 

.9 

2    20 

.1 

.2 

.3 

.5 

.7 

1.0 

2    30 

.1 

.2 

.3 

.5 

.8 

1.1 

DETERMINATION    OF   AZIMUTH. 

2  sin2  ^  T 
sin  1" 


151 


T 

0» 

1- 

2m 

3" 

4m 

5m 

6m 

~m 

8» 

I 

„ 

„ 

„ 

„ 

H 

„ 

„ 

It 

n 

0 

0.00 

1.96 

7.85 

17.67 

31.42 

49.09 

70.68 

96.20 

125.65 

1 

0.00 

2.03 

7.98 

17.87 

31.68 

49.41 

71.07 

96.66 

126.17 

2 

0.00 

2.10 

8.12 

18.07 

31.94 

49.74 

71.47 

97.12 

126.70 

3 

0.00 

2.16 

8.25 

18.27 

32.20 

50.07 

71.86 

97.58 

127.22 

4 

0.01 

2.23 

8.39 

18.47 

32.47 

50.40 

72.26 

98.04 

127.  75 

5 

0.01 

2.31 

8.52 

18.67 

32.74 

50.73 

72.66 

98.50 

128.28 

6 

0.02 

2.38 

8.66 

18.87 

33.01 

51.07 

73.06 

98.97 

128.  81 

7 

0.02 

2.45 

8.80 

19.07 

33.27 

51.40 

73.46 

99.43 

129.34 

g 

0.03 

2.52 

8.94 

19.28 

33.54 

51.74 

73.86 

99.90 

129.87 

9 

0.04 

2.60 

9.08 

19.48 

33.81 

52.07 

74.26 

100.37 

130.40 

10 

0.05 

2.67 

9.22 

19.69 

34.09 

52.41 

74.66 

100.84 

130.  94 

11 

0.06 

2.75 

9.36 

19.90 

34.36 

52.75 

75.06 

101.31 

131.  47 

12 

0.08 

2.83 

9.50 

20.11 

34.64 

53.09 

75.47 

101.  78 

132.  01 

13 

0.09 

2.91 

9.64 

20.32 

34.91 

53.43 

75.88 

102.25 

132.55 

14 

0.11 

2.99 

9.79 

20.53 

35.19 

53.77 

76.29 

102.  72 

133.09 

15 

0.12 

3.07 

9.94 

20.74 

35.46 

54.11 

76.69 

103.20 

133.63 

16 

0.14 

3.15 

10.09 

20.95 

35.74 

54.46 

77.10 

103.67 

134.  17 

17 

0.16 

3.23 

10.24 

21.16 

36.02 

54.80 

77.51 

104.15 

134.  71 

18 

0.18 

3.32 

10.39 

21.38 

36.30 

55.15 

77.93 

104.63 

135.25 

19 

0.20 

3.40 

10.54 

21.60 

36.58 

55.50 

78.34 

105.10 

135.80 

20 

0.22 

3.49 

10.69 

21.82 

36.87 

55.84 

78.75 

105.58 

136.  34 

21 

0.24 

3.58 

10.84 

22.03 

37.15 

56.19 

79.16 

106.06 

136.88 

22 

0.26 

3.67 

11.00 

22.25 

37.44 

56.55 

79.58 

106.55 

137.  43 

23 

0.28 

3.76 

11.15 

22.47 

37.72 

56.90 

80.00 

107.03 

137.  98 

24 

0.31 

3.85 

11.31 

22.70 

38.01 

57.25 

80.42 

107.51 

138.53 

25 

0.34 

3.94 

11.47 

22.92 

38.30 

57.60 

80.84 

107.99 

139.08 

26 

0.37 

4.03 

11.63 

23.14 

38.59 

57.96 

81.26 

108.48 

139.63 

27 

0.40 

4.12 

11.79 

23.37 

38.88 

58.32 

81.68 

108.  97 

140.18 

28 

0.43 

4.22 

11.95 

23.60 

39.17 

58.68 

82.10 

109.46 

140.74 

29 

0.46 

4.32 

12.11 

23.82 

39.46 

59.03 

82.52 

109.95 

141.29 

30 

0.49 

4.42 

12.27 

24.05 

39.76 

59.40 

82.95 

110.44 

141.85 

31 

0.52 

4.52 

12.43 

24.28 

40.05 

59.75 

83.38 

110.93 

142.40 

32 

0.56 

4.62 

12.60 

24.51 

40.35 

60.11 

83.81 

111.43 

142.  96 

33 

0.59 

4.72 

12.76 

24.74 

40.65 

60.47 

84.23 

111.92 

143.  52 

34 

0.63 

4.82 

12.93 

24.98 

40.95 

60.84 

84.66 

112.41 

144.08 

35 

0.67 

4.92 

13.10 

25.21 

41.25 

61.20 

85.09 

112.90 

144.  64 

36 

0.71 

5.03 

13.27 

25.45 

41.55 

61.57 

85.52 

113.40 

145.20 

37 

0.75 

5.13 

13.44 

25.68 

41.85 

61.94 

85.95 

113.90 

145.  76 

38 

0.79 

5.24 

13.62 

25.92 

42.15 

62.31 

86.39 

114.40 

146.33 

39 

0.83 

5.34 

13.79 

26.16 

42.45 

62.68 

86.82 

114.90 

146.89 

40 

0.87 

5.45 

13.96 

26.40 

42.76 

63.05 

87.26 

115.40 

147.46 

41 

0.91 

5.56 

14.13 

26.64 

43.06 

63.42 

87.70 

115.90 

14S.  03 

42 

0.96 

5.67 

14.31 

26.88 

43.37 

63.79 

88.14 

116.40 

148.60 

43 

1.01 

5.78 

14.49 

27.12 

43.68 

64.16 

88.57 

116.90 

149.  17 

44 

1.06 

5.90 

14.67 

27.37 

43.99 

64.54 

89.01 

117.41 

149.  74 

45 

.10 

6.01 

14.85 

27.61 

44.30 

64.91 

89.45 

117.92 

150.31 

46 

.15 

6.13 

15.03 

27.86 

44.61 

65.29 

89.89 

118.43 

150.88 

47 

.20 

6.24 

15.21 

28.10 

44.92 

65.67 

90.33 

118.  94 

151.45 

48 

.26 

6.36 

15.39 

28.35 

45.24 

66.05 

90.78 

119.  45 

152.03 

49 

.31 

6.48 

15.57 

28.60 

45.55 

66.43 

91.23 

119.96 

152.  61 

50 

.36 

6.60 

15.76 

28.85 

45.87 

66.81 

.91.68 

120.47 

153.19 

51 

.42 

6.72 

15.95 

29.10 

46.18 

67.19 

92.12 

120.98 

153.77 

52 

.48 

6.84 

16.14 

29.36 

46.50 

67.58 

92.57 

121.  49 

154.35 

53 

.53 

6.96 

16.32 

29.61 

46.82 

67.96 

93.02 

122.01 

154.93 

54 

.59 

7.09 

16.51 

29.86 

47.14 

68.35 

93.47 

122.53 

155.51 

55 

.65 

7.21 

16.70 

30.12 

47.46 

68.73 

93.92 

123.05 

156.09 

56 

.71 

7.34 

16.89 

30.38 

47.79 

69.12 

94.38 

123.57 

156.67 

57 

.77 

7.46 

17.08 

30.64 

48.11 

69.51 

94.83 

124.09 

157.25 

58 

.83 

7.60 

17.28 

30.90 

48.43 

69.90 

95.29 

124.61 

157.  84 

59 

.89 

7.72 

17.47 

31.16 

48.76 

70.29 

95.74 

125.  13 

158.  43 

i 

152 


U.  S.   COAST  AND  GEODETIC   SURVEY   SPECIAL  PUBLICATION   NO.   14. 

2  sin2  Yi  T 
sin  l'~ 


T 

9m 

10"' 

Urn 

12m 

13" 

14'" 

15" 

16" 

a 

„ 

„ 

„ 

„ 

„ 

,, 

„ 

„ 

0 

159.  02 

196.32 

237.54 

282.  68 

331.  74 

384.  74 

441.63 

502.  46 

1 

159.  61 

196.  97 

238.  26 

283.47 

332.59 

385.65 

442.62 

503.50 

2 

160.20 

197.63 

238.98 

284.26 

333.44 

386.56 

443.60 

504.55 

3 

160.80 

198.  28 

239.70 

285.04 

334.29 

387.  48 

444.58 

505.60 

4 

161.  39 

198.94 

240.42 

285.83 

335.  15 

388.40 

445.56 

506.65 

5 

161.  98 

199.60 

241.  14 

286.62 

336.00 

389.32 

446.55 

507.70 

6 

162.  58 

200.26 

241.  87 

287.41 

336.86 

390.24 

447.54 

508.76 

7 

163.17 

200.92 

242.60 

288.20 

337.72 

391.  16 

448.53 

509.81 

8 

163.  77 

201.59 

243.33 

289.00 

338.58 

392.09 

449.51 

510.86 

9 

164.37 

202.25 

244.08 

289.  79 

339.44 

393.01 

450.50 

511.92 

10 

164.97 

202.92 

244.79 

290.58 

340.30 

393.94 

451.50 

512.  98 

11 

165.57 

203.58 

245.52 

291.38 

341.  16 

394.86 

452.49 

514.  03 

12 

166.17 

204.25 

246.25 

292.18 

342.02 

395.  79 

453.48 

515.09 

13 

166.77 

204.92 

246.98 

292.98 

342.88 

3%.  72 

454.48 

516.  15 

14 

167.  37 

205.59 

247.  72 

293.78 

343.75 

397.65 

455.47 

517.  21 

15 

167.  97 

206.26 

248.45 

294.58 

344.62 

398.  58 

456.47 

518.  27 

16 

168.58 

206.93 

249.  19 

295.38 

345.  49 

399.52 

457.47 

519.  34 

17 

169.  19 

207.  60 

249.93 

296.18 

346.36 

400.45 

458.  47 

520.40 

18 

169.80 

208.27 

250.67 

296.99 

347.23 

401.38 

459.  47 

521.  47 

19 

170.  41 

208.94 

251.41 

297.79 

348.  10 

402.32 

460.47 

522.53 

20 

171.  02 

209.62 

252.15 

298.60 

348.  97 

403.26 

461.  47 

523.60 

21 

171.63 

210.  30 

252.89 

299.40 

349.  84 

404.20 

462.48 

524.  67 

22 

172.  24 

210.  98 

253.63 

300.21 

350.71 

405.14 

463.48 

525.74 

23 

172.85 

211.66 

254.37 

301.  02 

351.  58 

406.08 

464.48 

526.  81 

24 

173.47 

212.34 

255.12 

301.83 

352.46 

407.02 

465.49 

527.89 

25 

174.08 

213.  02 

255.87 

302.64 

353.34 

407.96 

466.50 

528.96 

26 

174.  70 

213.  70 

256.62 

303.46 

354.22 

408.90 

467.51 

530.03 

27 

175.  32 

214.  38 

257.37 

304.27 

355.10 

409.84 

468.52 

531.11 

28 

175.  94 

215.  07 

258.12 

305.09 

355.98 

410.  79 

469.53 

532.18 

29 

176.56 

215.  75 

258.87 

305.90 

356.86 

411.73 

470.54 

533.26 

30 

177.18 

216.44 

259.62 

306.72 

357.  74 

412.68 

471.55 

534.  33 

31 

177.80 

217.  12 

260.37 

307.54 

358.  62 

413.63 

472.  57 

535.41 

32 

178.43 

217.  81 

261.  12 

308.36 

359.  51 

414.  59 

473.58 

536.50 

33 

179.05 

218.50 

261.  88 

309.18 

360.39 

415.  54 

474.60 

537.  58 

34 

179.68 

219.  19 

262.64 

310.00 

361.  28 

416.  49 

475.  62 

538.  67 

35 

180.30 

219.88 

263.39 

310.  82 

362.  17 

417.44 

476.  64 

539.  75 

36 

180.93 

220.58 

264.15 

311.65 

363.07 

418.40 

477.65 

540.83 

37 

181.56 

221.27 

264.91 

312.  47 

363.96 

419.  35 

478.  67 

541.91 

38 

182.  19 

221.97 

265.68 

313.  30 

364.85 

420.31 

479.  70 

543.00 

39 

182.82 

222.66 

266.44 

314.  12 

365.75 

421.27 

480.72 

544.09 

40 

183.46 

223.36 

267.20 

314.  95 

366.64 

422.23 

481.  74 

545.18 

41 

184.09 

224.06 

267.96 

315.  78 

367.53 

423.19 

482.77 

546.27 

42 

184.72 

224.76 

268.73 

316.  61 

368.  42 

424.15 

483.79 

547.  36 

43 

185.35 

225.46 

269.49 

317.44 

369.31 

425.11 

484.82 

548.  45 

44 

185.99 

226.16 

270.26 

318.  27 

370.  21 

426.07 

485.85 

549.55 

45 

186.63 

226.86 

271.02 

319.  10 

371.  11 

427.04 

486.88 

550.64 

46 

187.27 

227.57 

271.  79 

319.94 

372.  01 

428.  01 

487.91 

551.73 

47 

187.  91 

228.27 

272.  56 

320.  78 

372.  91 

428.97 

488.94 

552.83 

48 

188.55 

228.  98 

273.34 

321.  62 

373.82 

429.93 

489.97 

553.93 

49 

189.19 

229.68 

274.  11 

322.45 

374.  72 

430.90 

491.01 

55.5.  03 

50 

189.83 

230.39 

274.88 

323.29 

375.  62 

431.87 

492.05 

556.  13 

51 

190.47 

231.10 

275.65 

324.  13 

376.  52 

432.84 

493.  08 

557.  24 

52 

191.  12 

231.81 

276.43 

324.97 

377.43 

433.  82 

494.12 

558.34 

53 

191.  76 

232.52 

277.20 

325.81 

378.  34 

434.  79 

495.  15 

559.44 

54 

192.  41 

233.24 

277.98 

326.66 

379.  26 

435.76 

496.19 

560.55 

55 

193.06 

233.95 

278.76 

327.50 

380.17 

436.73 

497.23 

561.65 

56 

193.71 

234.67 

279.55 

328.35 

381.08 

437.71 

498.  28 

562.76 

57 

194.  36 

235.38 

280.33 

329.19 

381.99 

438.69 

499.32 

563.87 

58 

195.01 

236.10 

281.12 

330.04 

382.90 

439.67 

500.37 

564.98 

59 

195.66 

236.82 

281.90 

330.89 

383.82 

440.65 

501.41 

566.08 

DETERMINATION   OF   AZIMUTH. 


153 


METHOD  OF  REPETITIONS— EXAMPLE  OF  RECORD  AND  COMPUTATION. 

Remarks  similar  to  those  appearing  on  page  145  apply  here  also.  The  observations  required 
to  determine  the  azimuth  of  a  mark  by  the  method  of  repetitions  are  the  same  as  those  required 
to  measure  a  horizontal  angle  in  a  triangulation  with  the  same  repeating  theodolite,  with  the 
addition  of  level  readings,  and  readings  of  the  chronometer  at  the  instants  of  the  pointings 
upon  the  star. 

The  adjustments  required  are  those  mentioned  on  page  145,  with  the  exception  that  a 
repeating  theodolite  is  ordinarily  read  by  verniers  instead  of  microscopes. 

Record — Azimuth  by  repetitions. 

[Station,  Kahatchee  A.    State,  Alabama.    Date,  June  6, 1898.    Observer,  O.  B.  F.    Instrument,  10-inch  Gambey  No.  63.    Star,  Polaris.] 

[One  division  striding  level=2".67.] 


Objects 

Chr.  time  on 
star 

Pos.  of 
tel. 

Repeti- 
tions 

Level  read- 
ings 

W             E 

Circle  readings 

Angle 

i 

1 

A 

n 

B 

Mean 

Mark 

D 

0 

178 

03 

22.5 

20 

21.2 

Star 

14h  46m  30' 

1 

4.  5     10.  7 
9.  2      5.  9 

49     OS 

2 

52    51 

D 

3 

9.  6      5.  6 
5.  2    10.  0 

56    10 

R 

4 

11.3      4.0 

7.  8      7.  4 

Set  No.  5 

14   59     12 

5 

15  01    55 

R 

6 

8.  7      6.  6 
11.  9      3.  4 

100 

16 

20 

20 

20 

72'  57'  50".  2 

]4   54     17.7 

68.  2    53.  6 

+  14.6 

Star 

15  04    44 

R 

1 

11.9      3.4 

8.  5      6.  8 

07     18 

'2 

09    54 

R 

3 

7.9      7.3 
11.  2      4.  1 

Set  No.  6 

14    15 

D 

4 

9.  0      6.  1 
5.  9      9,  6 

16    14 

5 

15   18    24 

6 

5.  9      9.  6 
9.  1      6.  2 

Mark 

D 

177 

27 

00 

00 

00 

72°  51'  46".  7 

15   11    48.2 

69.  4    53.  1 

+16.3 

i 

154 


U.   S.    COAST   AND   GEODETIC   SUKVEY   SPECIAL   PUBLICATION    NO.    14. 

Computation — Azimuth  by  repetitions. 

[Kahatchee,  Ala.    ^-33°  13'  40".33.] 


Date,  1898,  set 

June  6             5 

June  6           6 

Chronometer  reading 

14    54    17.  7 

15    11    48.  2 

Chronometer  correction 

-31.1 

-31.1 

Sidereal  time 

14     53    46.  6 

15    11     17.1 

noi  Polaris 

1    21    20.  3 

1    21    20.  3 

t  of  Polaris  (time) 

13    32    26.  3 

13    49    56.  8 

t  of  Polaris  (arc) 

203°  06'  34".  5 

207°  29'  12".  0 

d  of  Polaris 

88    45      46.  9 

log  cot  S 

8.  33430 

8.  33430 

log  tan  <j> 

9.  81629 

9.  81629 

log  cos  t 

9.  96367n 

9.  9479871 

log  a  (to  five  places) 

8.  11426n 

8.  09857n 

log  cot  3 

8.  334305 

8.  334305 

log  sec  <j> 

0.  077535 

0.  077535 

log  sin  t 

9.  593830w 

9.66421171 

log  q  

9.  994387 

9.  994584 

"  1—  a 

log  (  —  tan  ^4)  (to  6  places) 

8.  00005771 

8.  070635?i 

.A=Azimuth  of  Polaris,  from 

north* 

0°  34'  22".  8 

0°  40'  26".  8 

m       s           " 

TO         *              " 

[1    47.7     119.3 

7     04.2     98.1 

5    09.  7      52.  3 

4    30.  2    39.  8 

2sin2J  T 

1     26.  7        4.  1 

1     54.  2       7.  1 

T  ana     gjn  ^// 

1    52.  3        6.  9 

2    26.8    11.8 

4     54.  3      47.  2 

4    25.8    38.5 

7     37.3     114.0 

6    35.  8    85.  4 

Sum 

343.8 

280.7 

Mean 

57.3 

46.8 

1  r2  sin  2J  r 

1    7C>8 

1.  670 

log  if    sin  1" 

-L.    t  <JO 

log  (curvature  corr.) 

9.758 

9.741 

Curvature  correction 

-0.6 

-0.6 

Altitude  of  Polaris=A 

32°  07' 

-T  tan  A=level  factor 

.419 

.419 

Inclination  t 

+3.6 

+4.1 

Level  correction 

—1".  5 

-1".  7 

Angle,  star  —  mark 

72    57    50.  2 

72     51     46.  7 

Corrected  angle 

72    57    48.  7 

72    51    45.  0 

Corrected  azimuth  of  star* 

0    34    22.  2 

0    40    26.  2 

Azimuth  of  mark  E  of  N 

73     32     10.  9 

73    32    11.  2 

180     00    00.  0 

180    00    00.  0 

Azimuth  of  mark 

253     32     10.  9 

253    32    11.  2 

(Clockwise  from  south) 

To  the  mean  result  from  the  above  computation  must  be  applied  corrections  for  diurnal  aberration  and  eccentricity  (if  any)  of  Mark.    Carry 
times  and  angles  to  tenths  of  seconds  only. 

*  Minus  if  west  of  north.  t  See  footnote  on  p.  148. 


DETERMINATION   OF   AZIMUTH.  155 

METHOD    OF    REPETITIONS—  EXPLANATION    OF    RECORD    AND    COMPUTATION. 

Throughout  the  observations  the  instrument  was  always  turned  in  a  clockwise  direction 
about  its  vertical  axis.  In  set  No.  5  the  swing  from  the  mark  to  the  star  was  made  with  the 
upper  motion  loose  and  lower  motion  clamped,  and  therefore  with  the  circle  reading  changing, 
and  in  set  No.  6  the  reverse  was  the  case.  In  set  No.  5  the  explement  of  the  small  angle  between 
the  star  and  the  mark  was  really  measured,  while  in  No.  6  the  angle  itself  was  measured.  Both 
results  may  be  computed  directly  in  terras  of  the  angle  by  making  the  subtractions  thus,  in  set 

No.  5. 

,       (360°  +  178°  03'  21//.2)-100°  16'  20".Q  ,       „ 

angle  =—  fi  =72     57    50    .2 

in  set  No.  6, 

,       (3600  +  177°27/00//.0)-1000  16'  20".  0     790  «/  x«//  71 
angle  =  —  —  ^  —  =72"  51    4o    .7  . 

If  the  clamp  on  the  horizontal  circle  produces  a  constant  error,  either  by  dragging  or 
overrunning,  these  two  results  will  be  equally  in  error  with  opposite  signs,  and  their  mean  will 
be  free  from  the  constant  part  of  the  clamp  error.  Hence,  it  is  desirable  to  observe  the  sets 
alternately  in  the  order  Mark-Star,  Star-Mark,  as  here  indicated. 

The  summary  of  results  for  this  station  shows  37  sets  of  observations  were  made  on  four 
nights.  From  the  18  sets  observed  in  the  order  Star-Mark  the  mean  azimuth  was  73°  32'  12".07, 
and  from  the  19  sets  observed  in  the  order  Mark-Star  the  mean  was  73°  32'  12".89,  showing 
that  the  clamp  error  was  very  small.  The  adopted  indiscriminate  mean  of  all  the  37  sets  was 
73°  32'  12".49.  The  correction  for  diurnal  aberration  (  +  0".31)  being  applied,  the  resulting 
azimuth  of  the  mark,  E.  of  N.  equals  73°  32'  12".80±0".16.  The  probable  error  of  a  single 


0.455  „  QS 

n_1} 

During  these  observations  the  instrument  was  supported  upon  its  tripod,  the  legs  of  which 
were  set  upon  large  stakes  driven  solidly  into  the  ground. 

The  level  readings  were  taken  with  the  first,  third,  fourth,  and  sixth  pointings  upon  the 
star,  that  is,  at  the  beginning  and  end  of  the  set  and  just  before  and  just  after  the  reversal  of 
the  telescope.  In  each  case  the  level  was  read  in  one  position  just  before  perfecting  the  pointing 
upon  the  star,  and  in  the  other  position  immediately  after  the  pointing  upon  the  star.  The 
value  of  one  division  of  the  level  was  2".67. 

The  computation  needs  no  further  explanation.     The  formula 


tan  A  =  —  cot  d  sec  <p  sin  t  (    _    } 


was  used. 

The  correction  for  elevation  of  mark,  when  appreciable,  is  applied  in  the  final  summary 
of  results,  just  as  in  the  case  of  the  direction  method.  The  reduction  to  the  mean  position  of  the 
pole  is  also  applied  to  the  final  result,  but  for  observations  previous  to  the  year  1900  no  such 
reduction  can  now  be  made.  (See  p.  85.) 

MICROMETRIC  METHOD— EXAMPLE  OF  RECORD  AND  COMPUTATION. 

In  the  micrometric  method 2  the  small  difference  of  azimuth  of  the  star  and  the  mark  is 
measured  with  an  eyepiece  micrometer,  independently  of  the  graduated  horizontal  circle  of 
the  instrument,  even  if  it  has  one.  The  mark  must  therefore  be  placed  nearly  in  the  vertical 
of  the  star  at  the  time  at  which  it  is  to  be  observed.  The  method  may  be  used  with  the  star  at 
any  hour-angle,  but  unless  the  star  is  near  elongation  it  will  pass  beyond  the  safe  range  of  the 
micrometer  after  but  two  or  three  sets  of  observations  have  been  taken,  whereas  if  the  mark 

1  The  computer  should  notice  the  convenient  fact  that  in  dividing  an  angle  by  six  the  remainder,  when  the  degrees  are  divided,  is  the  tens 
figure  in  the  minutes,  and  the  remainder  in  the  minutes  is  the  tens  figure  in  the  seconds. 

*  For  an  account  of  this  method,  together  with  some  historical  notes,  see  Appendix  No.  2  of  the  Report  for  1891. 


156 


U.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.    H. 


is  placed  nearly  under  the  star  at  elongation  (preferably  one  or  two  minutes  of  arc  inside)  the 
observations  may  be  continued  for  two  hours  or  more  and  the  results  will  also  be  nearly  inde- 
pendent of  the  chronometer  error.  The  instrument  used  may  be  a  theodolite,  a  meridian 
telescope,  a  transit,  or,  in  fact,  any  instrument  having  a  stable  horizontal  axis  and  furnished 
with  an  eyepiece  micrometer  capable  of  measuring  angles  in  the  plane  defined  by  the  telescope 
and  its  horizontal  axis. 

Record  and  computation — Azimuth  ~by  micrometric  method. 

[Station  No.  10,  Mexican  Boundary.    Date,  October  13,  1892.    Observer,  J.  F.  H.    Instrument,  Fauth  Repeating  Theodolite,  No.  725  (10  in.). 

Star,  Polaris  near  eastern  elongation.) 


Circle 

Level  readings 
W            E 

Chronometer 
time 

T 

2  sin  !  $  T 

Micrometer  readings  — 

sin  1" 

On  star 

On  mark 

E 

E 

W 

w 

8.  0        9.  9 
10.  0        7.  3 

h    m     s 
9  06  38.  0 
07  32.  0 

08  05.  5 
09  13.0 
09  48.  0 

9  12  01.  8 
12  24.  7 

12  48.  3 
13  36.  3 
13  58.  1 

m    s 
3  58.6 
3  04.6 

2  31.  1 
1  23.6 
0  48.6 

1  25.2 
1  48.1 

2  11.7 
2  59.7 
3  21.5 

31.  05 
18.59 

12.45 
3.82 
1.29 

3.96 
6.37 

9.46 

17.61 
22.  11 

18'.  379 
.388 

.400 
.424 
.430 

18'.  310 
.315 

.315 
.311 
.316 

;.=2»  12m  W.  of 
Washington 

A.      01  o   -in/   oc// 

+  18.0  -17.2 
+0.8 

9.  0      9.  0 
7.0    10.9 

1   div.   of  level 
=3".68 

1    turn   of  mic. 

m//  70 

18.  4042 

18.  100 
.100 

.090 
.086 
.080 

18.3134 

18.  290 
.275 

.279 
.281 
.279 

Means 
Means 

+16.0-19.9 
—  3  9 
Mean        ld.  55 

9  10  36.  6 

12.67 

18.  0912 

18.  2808 

£  of  star  at  middle  of  first  half  of  set=58°  48'. 
£  of  star  at  middle  of  second  half  of  set=58°  46'. 
a=lh  20m  07-.4. 


cosec  C=1.1691 .     cot  58°  47'=0.  606. 
cosec  C=1.1695. 
<J=88°  44'  10". 4. 


Collimadon  axis  raads  4  (18.3134+18.2808)1  =18t.2971 

Mark  east  of  colHmation  axis  18.3134-18. 2971  =0.0163=     02".02 

Circle  E.,  star  E.  of  collimation  axis  (18.4042-18.2971)  (1.1691)=  0  .1252 

Circle  W.,  star  E.  of  collimation  axis  (18.2971-18.0912)  (1.1695)=  0  .2408 


Mean,  star  E.  of  collimation  axis 

Mark  west  of  star 
Level  correction  (]  .55)  (0.92)  (0.606) 


=  0  .1830=     22   .64 


=     20   .62 
=  -  0   .86 


Mark  west  of  star,  corrected  =     19   .76 

Mean  chronometer  time  of  observation=     21h  10m  36S.6 
Chronometer  coirection  =—2    11    28  .2 

Sidereal  time  =    18    59    08  .4 

a  =      1    20    07  .4 


log.  cot  S 
log.  tan  <j> 
log.  cos  t 

log.  a 


Hour-angle,  t,  in  time 

in  arc 

=  8. 34362 
=  9. 78436 
=  8.  96108  n 


17    39    01  .0 
264°  45'  15".0 


=     7.  08906  n 


1  In  this  instrument  increased  readings  of  the  micrometer  correspond  to  a  movement  of  the  line  of  sight  toward  the  east  when  the  vertical 
circle  is  to  the  east,  and  toward  the  west  when  the  vertical  circle  is  to  the  west. 


DETERMINATION   OF   AZIMUTH.  1.57 


log.  cot  8  =     8. 343618 

log.  sec  ^  =0. 068431 

log.  &in  t  =    9.  998177  n 

loe.  5 — -  =     9.  999467 

"       1 — I* 


1  g.  (-tan  4)  =    8.  409693  n 

A  =+1°  28'  16".91 

log.  12.67  =     1. 10278 

log.  curvature  corr.  =     9. 51247 

Curvature  corr.  =  —0. 33 

Diur.  Aber.  corr.  =  +0.  32 


Mean  azimuth  of  star          =  +  1°  28'  16".90 
Mark  west  of  star  19  .  76 


Azimuth  of  mark,  E.  of  N.=+l°  27'  57",14 

The  correction  for  elevation  of  mark  and  the  reduction  to  the  mean  position  of  the  pole 
are  applied  to  the  final  result  of  the  separate  measures  at  a  station.  In  the  case  of  this  par- 
ticular station  the  necessary  information  is  not  yet  available  for  reduction  to  the  mean  position 
of  the  pole.  (See  p.  85.) 

MICROMETRIC    METHOD — EXPLANATION    OF    RECORD    AND    COMPUTATION. 

The  compact  form  of  record  shown  above  does  not  indicate  the  order  in  which  the  obser- 
vations were  taken.  The  micrometer  line  is  placed  nearly  in  the  collimation  axis  of  the  tele- 
scope, a  pointing  made  upon  the  mark  by  turning  the  horizontal  circle,  and  the  instrument  is 
then  clamped  in  azimuth.  The  program  is  then  to  take  five  pointings  upon  the  mark;  direct 
the  telescope  to  the  star;  place  the  striding  level  in  position;  take  three  pointings  upon  the 
star  with  chronometer  times;  read  and  reverse  the  striding  level;  take  two  more  pointings 
upon  the  star,  noting  the  times;  read  the  striding  level.  This  completes  a  half -set.  The  hori- 
zontal axis  of  the  telescope  is  then  reversed  in  its  Y's;  the  telescope  pointed  approximately  to 
the  star;  the  striding  level  placed  in  position;  three  pointings  taken  upon  the  star  with  observed 
chronometer  times;  the  striding  level  is  read  and  reversed;  two  more  pointings  are  taken  upon 
the  star,  with  observed  times;  the  striding  level  is  read,  and  finally  five  pointings  upon  the 
mark  are  taken. 

Three  such  complete  sets  may  be  observed  in  from  thirty  to  fifty  minutes.  The  effect  of  a 
uniform  twisting  of  the  instrument  in,  azimuth  is  eliminated  from  the  result.  The  bubble  of 
the  striding  level  has  plenty  of  time  to  settle  without  delaying  the  observer  an  instant  for  that 
purpose. 

The  zenith  distance  of  the  star  should  be  read  occasionally,  once  during  each  set,  say,  as  it 
is  needed  in  making  the  computation.  If  it  is  read  with  one  of  the  star  pointings  in  each  set, 
its  value  at  any  other  time  may  be  obtained  with  sufficient  accuracy  by  interpolation. 

It  should  be  borne  in  mind  in  making  the  computation  that  the  micrometer  measures 
angles  in  the  plane  defined  by  the  telescope  and  its  horizontal  axis.  To  reduce  the  measured 
angle  between  the  collimation  axis  and  the  star  to  a  horizontal  angle,  it  must  be  multiplied  by 
cosec  £,  as  indicated  in  the  computation.  To  avoid  ah1  approximation  in  the  computation  it 
would  be  necessary  to  reduce  each  pointing  upon  the  star  separately,  as  the  zenith  distance  is 
constantly  changing.  It  is  sufficiently  accurate,  however,  to  reduce  the  mean  of  the  pointings 
of  a  half-set  with  the  mean  zenith  distance  of  that  half-set,  as  indicated  in  the  computation. 
To  use  a  single  zenith  distance  for  the  whole  set  will  sometimes  introduce  errors  which  are  rather 
too  large  to  be  neglected.  The  factor  cosec  £  will  not,  in  general,  be  necessary  in  connection  with 
pointings  upon  the  mark,  because  the  mark  will  usually  be  nearly  in  the  horizon  of  the  instru- 
ment, and  cosec  £  therefore  nearly  unity,  and  because  the  collimation  axis  is  purposely  placed 
as  nearly  as  possible  upon  the  mark  and  the  angle  concerned  is  therefore  very  small. 

The  micrometer  value  may  be  determined  by  observations  upon  a  star  near  culmination 
by  the  process  outlined  on  page  124.  If  the  striding  level  is  read  in  connection  with  such  obser- 


158  U.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.   14. 

vations,  the  correction  to  be  applied  to  each  observed  time  to  reduce  it  to  what  it  would  have 
been  with  the  transverse  axis  horizontal  is 

..  1  dcos  rsec  d 

- 


for  upper  culmination  and  for  a  level  of  which  the  graduation  is  numbered  both  ways  from  the 
middle.  For  lower  culmination  the  sign  of  the  correction  must  be  reversed. 

Another  convenient  way  of  determining  the  micrometer  value,  all  in  daylight,  is  to  measure 
a  small  horizontal  angle  at  the  instrument  between  two  terrestrial  objects,  both  with  the 
micrometer  and  the  horizontal  circle  of  the  theodolite.  This  method  is  less  liable  to  constant 
errors  than  the  circumpolar  method. 

If  the  azimuth  mark  is  placed  to  the  southward  of  the  station,  the  program  of  observing 
and  the  computation  are  but  slightly  modified. 

DISCUSSION  OF  ERRORS. 

It  is  convenient  and  conducive  to  conciseness  to  discuss  separately  the  external  errors, 
observer's  errors,  and  instrumental  errors,  which  together  comprise  the  errors  of  observation. 

The  external  errors  affecting  azimuth  determinations  are  those  due  to  lateral  refraction 
of  the  rays  of  light  from  the  star  or  mark  to  the  instrument,  to  errors  in  the  adopted  right 
ascension  and  declination  of  the  star  observed,  and  to  error  in  the  adopted  latitude  of  the  sta- 
tion of  observation. 

Examination  of  many  series  of  azimuth  observations  indicates  that,  in  general,  they  are 
subject  to  some  error  which  is  peculiar  to  each  night  of  observation,  and  constant  for  that 
night,  but  changes  from  night  to  night.  For  example,  from  144  sets  of  micromctric  observa- 
tions of  azimuth,  made  on  36  different  nights  at  15  stations  on  the  Mexican  boundary  in 
1892-93,  it  was  found  that  the  error  peculiar  to  each  night  was  represented  by  the  probable 
error  ±0".38,  and  that  the  probable  error  of  each  set,  exclusive  of  this  error,  was  ±0".54.1 
In  other  words,  in  this  series  of  observations  the  error  peculiar  to  each  night,  which  could  not 
have  been  eliminated  by  increasing  the  number  of  observations  on  that  night,  was  two-thirds 
as  large,  on  an  average,  as  the  error  of  observation  in  the  result  from  a  single  set.  Similarly, 
from  the  published  results  of  418  sets  of  micrometric  observations  on  8  nights  at  a  European 
station,2  it  follows  that  the  error  peculiar  to  each  night  was  ±0".55,  while  the  probable  error 
of  a  single  set  was  ±0".80.  The  micrometric  observations  are  peculiarly  adapted  to  exhibiting 
this  error,  because  of  their  great  accuracy  and  the  rapidity  with  which  they  may  be  taken. 
Azimuth  was  observed  at  73  stations  on  the  transcontinental  triangulation  along  the  thirty- 
ninth  parallel.  Most  of  these  observations  were  taken  by  the  direction  method,  and  although 
they  are,  for  various  reasons,  but  poorly  adapted,  as  a  rule,  to  exhibiting  the  errors  peculiar 
to  the  separate  nights,  there  are  no  less  than  16  cases  out  of  the  73  in  which  a  mere  inspection 
indicates  that  there  were  errors  of  that  character. 

The  most  plausible  explanation  of  the  above  facts  seems  to  be  that  there  is  lateral  refrac- 
tion between  the  mark  and  the  instrument,  dependent  upon  the  peculiar  atmospheric  condi- 
tions of  each  night.  Whether  that  explanation  be  true  or  not,  the  fact  remains  that  an  increase 
of  accuracy  in  an  azimuth  determination  at  a  given  station  may  be  attained  much  more  readily 
by  increasing  the  number  of  nights  of  observation  than  by  increasing  the  number  of  sets  on 
each  night.  If  one  series  of  observations  is  made  early  in  the  evening  and  another  series  just 
before  dawn  on  the  same  night,  these  series  may  be  considered,  in  so  far  as  the  preceding  sen- 
tence is  concerned,  to  be  on  different  nights,  as  the  atmospheric  conditions  will  have  been 
given  an  opportunity  to  change. 

The  line  from  the  station  to  the  mark  should  not  pass  close  to  any  objects,  such  as  a  smoke- 
stack, building,  clump  of  trees,  or  a  hill.  Even  when  the  line  is  close  to  the  ground  which  has 

1  See  Report  of  International  Boundary  Commission,  United  States  and  Mexico,  1891-96  (Washington,  1898),  pp.  69-72. 
1  Station  Kampenwand.    See  pp.  68-92,  Veroflentlichung  der  Konigl.  Bayerischen  Commission  Jiir  die  Internationale  Erdmessung,  Astron. 
omische-Geodatische  Arbeiten,  Heft  2  (Miinchen,  1897). 


DETERMINATION   OF   AZIMUTH.  159 

a  decided  slope  normal  to  the  line,  there  may  be  decided  lateral  refraction.  During  the  primary 
triangulation  in  the  city  of  Greater  New  York  the  errors  on  the  lines  which  were  close  to  stacks 
and  buildings  were  much  greater  than  on  the  clear  lines.  There  was  a  line  in  the  Texas-Cali- 
fornia arc  of  primary  triangulation  which  at  one  point  was  very  close  to  the  side  of  a  steep 
hill.  The  line  was  observed  from  the  end  nearest  the  hill  on  several  days  and  nights,  with  a 
total  range  in  the  means  for  the  several  observing  periods  of  7.7  seconds  of  arc.  It  was  found 
that  the  observations  made  when  the  wind  was  blowing  across  the  line  toward  the  hill  gave 
the  more  reliable  results.  (See  p.  62  of  Special  Publication  No.  11  of  the  U.  S.  Coast  and  Geo- 
detic Survey.) 

The  positions  of  the  four  principal  close  circumpolars  have  been  determined  by  so  manjr 
observations  at  the  fixed  observatories  under  such  favorable  conditions  that  it  is  difficult  to 
believe  that  the  errors  in  their  adopted  right  ascensions  and  decimations  are  sufficiently  large  to 
produce  errors  in  the  computed  azimuths  that  are  otherwise  than  small  in  comparison  with  the 
other  errors  involved  in  the  azimuth  observations.  On  the  other  hand,  when  Polaris  (or  some 
other  circumpolar)  has  been  observed  at  both  culminations  or  both  elongations,  at  a  given 
station,  the  observations  at  one  culmination  (or  elongation)  have  often  shown  a  tendency  to 
differ  by  a  constant  from  those  at  the  other  culmination  (or  elongation),  as  if  the  adopted  right 
ascension  (or  declination)  were  in  error.  It  should  be  borne  in  mind  in  such  cases  that  the 
atmospheric  conditions  have  been  reversed,  so  to  speak,  between  the  culminations  (or  elonga- 
tions) ;  for  in  one  case  the  temperature  will  be  rising  and  in  the  other  falling,  in  general,  the 
two  cases  occurring  at  the  extreme  ends  of  darkness  or  of  light,  or  one  in  the  darkness  and  the 
other  in  the  light.  Hence  only  a  long  and  careful  investigation  will  determine  whether  such 
constant  differences  are  due  to  defective  star  places  or  to  changed  atmospheric  conditions. 
It  is  important  from  a  practical  point  of  view  to  note  that  if  the  azimuth  observations  at  a 
station  are  all  made  upon  one  star  and  are  equally  distributed  between  two  hour-angles,  differ- 
ing by  about  twelve  hours,  the  mean  result  will  be  sensibly  independent  of  the  errors  of  the 
adopted  right  ascension  and  declination,  and  the  conditions  will  be  decidedly  favorable  to 
eliminating  the  effects  of  lateral  refraction  from  the  mean  result. 

An  error  in  the  adopted  latitude  of  the  station  produces  the  maximum  effect  when  the  star 
is  observed  at  elongation  and  is  without  effect  if  the  star  is  observed  at  culmination.  For 
Polaris  at  elongation,  to  produce  an  error  of  0".01  in  the  computed  azimuth  the  adopted  lati- 
tude must  be  in  error  by  0".70  for  a  station  in  latitude  30°,  and  by  0".14  for  a  station  in  latitude 
60°.  The  error  in  the  computed  azimuth  from  this  source  will  be  larger  for  a  star  farther  from 
the  pole.  The  astronomic  latitude  (defined  by  the  actual  line  of  gravity  at  the  station)  is 
required  for  the  computation,  and  not  the  geodetic  latitude.  This  error,  which  will  in  general 
be  very  small,  will  also  be  eliminated  by  observing  the  star  at  two  positions  about  twelve  hours 
apart. 

The  observer's  errors  are  his  errors  of  pointing  upon  the  mark  and  star,  errors  of  pointing 
upon  the  circle  graduation  if  reading  microscopes  are  used,  errors  of  vernier  reading  if  verniers 
are  used,  errors  of  reading  the  micrometer  heads,  errors  in  reading  the  striding  level,  and  errors 
in  estimating  the  times  of  the  star  pointings.  There  is  such  a  large  range  of  difference  in  the 
designs  of  the  various  instruments  used  for  azimuth  observations  that  little  can  be  said  of  the 
relative  and  absolute  magnitude  of  these  errors  that  will  be  of  general  application.  Each 
observer  should  investigate  these  errors  for  himself  with  the  particular  instrument  in  hand.  It 
will  be  found  in  general  that  the  observer's  errors  play  a  minor  part  in  furnishing  the  final 
errors  of  the  results,  except  perhaps  in  the  micrometric  method. 

The  effect  of  errors  in  tune,  either  errors  in  estimating  the  times  of  the  star  pointings,  the 
personal  equation  of  the  observer,  or  errors  in  the  adopted  chronometer  correction,  may  be 
estimated  by  noting  the  rate  at  which  the  star  was  moving  in  azimuth  when  the  observations 
were  made.  Such  errors  are  usually  small,  but  not  insensible  except  near  elongation,  and  will 
tend  to  be  eliminated  by  observations  of  the  same  star  at  two  hour-angles  differing  by  about 
twelve  hours. 


160  U.   S.   COAST  AND  GEODETIC   SURVEY  SPECIAL  PUBLICATION   NO.   14. 

Of  the  magnitude  of  the  instrumental  errors  arising  from  imperfect  adjustment  and  imperfect 
construction  and  imperfect  stability  little  of  general  application  can  be  said,  because  of  the 
great  variety  of  the  instruments. 

With  the  larger  and  more  powerful  instruments  the  errors  due  to  instability  become  rela- 
tively great  and  should  be  guarded  against  by  careful  manipulation  and  rapid  observing,  by 
using  a  carefully  planned  program  of  observations,  and  by  protecting  the  instrument  against 
temperature  changes  as  far  as  possible.  In  this  connection  it  should  be  noted  that  each  of  the 
programs  of  observation  given  on  the  preceding  pages  is  especially  adapted  to  elimination  of 
the  effect  of  any  continuous  twisting  of  the  instrument  in  azimuth,  and  is  so  planned  that  the 
observer  will  not  ordinarily  lose  time  in  waiting  for  the  bubble  of  the  striding  level  to  come  to 
rest.  That  observer  of  azimuth  will  be  most  successful  in  avoiding  errors  due  to  instability 
who  keeps  it  most  clearly  and  continuously  in  mind  that  the  instrument  and  its  support  are 
made  of  elastic  material  of  such  a  large  coefficient  of  thermal  expansion  that  no  part  remains 
of  fixed  dimensions  or  shape.  He  will  be  especially  careful  about  the  thermal  conditions  and 
the  stresses  to  which  his  instrument  is  subjected  and  will  observe  with  the  greatest  rapidity 
consistent  with  allowable  observer's  errors. 

The  errors  due  to  the  striding  level  become  more  serious  the  farther  north  is  the  station,  as 
may  be  seen  by  inspection  of  the  formula  for  the  level  correction  (p.  144). 

The  errors  of  graduation  of  the  horizontal  circle  have  the  same  effect  in  azimuth  observa- 
tions as  in  observations  of  horizontal  angles.  The  number  of  positions  in  which  the  circle  must 
be  used  in  the  direction  method  may  therefore  be  decided  upon  the  same  basis  as  in  the  angle 
measurements. 

The  micrometric  method  gives  a  higher  degree  of  accuracy  than  either  the  method  of 
repetitions  or  the  method  of  directions.  This  is  probably  due  largely  to  the  great  rapidity  with 
which  the  observations  may  be  made,  a  condition  which  is  very  favorable  to  the  elimination  of 
errors  due  to  instability  of  the  instrument  and  its  support.  The  error,  in  the  final  result  for  a 
station  by  this  method,  due  to  an  error  in  the  adopted  value  of  one  turn  of  the  micrometer  may 
be  made  very  small  by  so  placing  the  azimuth  mark  (or  marks)  and  so  timing  the  observations 
that  the  sum  of  the  angles  measured  eastward  from  the  mark  (or  marks)  to  the  star  shall  be 
nearly  equal  to  the  sum  of  such  angles  measured  westward. 

STATEMENT  OF  COSTS. 

When  azimuths  are  observed  with  a  theodolite  during  the  progress  of  a  triangulation  the 
cost  is  very  small.  This  is  the  method  now  employed  in  the  primary  triangulation  by  the  Coast 
and  Geodetic  Survey.  It  is  probable  that  the  observations  and  field  computations  for  an 
azimuth  do  not  involve  an  additional  cost  of  more  than  $50  per  azimuth  station. 

If,  however,  the  azimuths  are  observed  by  a  separate  party  some  time  later  than  the  tri- 
angulation, and  when  there  is  more  or  less  building  of  signals  at  the  stations  at  each  end  of  the 
line  for  which  the  azimuth  is  determined,  the  cost  per  station  will  vary  during  a  season's  opera- 
tions from  $200  to  $500.  When  an  observer  must  go  out  in  the  field  to  determine  a  single 
azimuth  at  a  distant  point  the  expense  may  be  more  than  $500.  A  season's  work  should  be 
planned  so  that  the  cost  and  time  of  traveling  between  stations  will  be  a  minimum.  If  prac- 
ticable, the  work  in  any  locality  should  be  done  at  that  time  of  the  year  when  the  most  favorable 
weather  conditions  may  be  expected. 

AZIMUTH   FROM  TIME   OBSERVATIONS. 

For  a  number  of  years  azimuths  of  a  secondary  degree  of  accuracy  for  use  in  connection 
with  tertiary  triangulation  in  Alaska  have  been  obtained  directly  from  time  observations  with 
a  transit  or  meridian  telescope,  with  little  additional  labor  of  observing  and  computing.  With 
the  adoption  of  the  transit  micrometer  the  accuracy  of  the  results  was  greatly  increased, 
approaching  primary  in  character.  This  method  of  determining  azimuths  has  proved  of  great 
value  in  high  latitudes  where  slow-moving  stars  are  high  in  altitude,  and,  consequently,  satis- 
factory azimuths  from  observations  with  a  theodolite  are  difficult  to  obtain. 


DETERMINATION    OF   AZIMUTH.  161 

Observations  on  a  mark  which  is  set  closely  in  the  meridian  are  made  during  each  half 
set  of  observations  for  time.  See  page  80  for  description  of  method  of  observing  time  in 
high  latitudes.  The  azimuth  correction,  computed  from  the  time  observations,  is  combined 
with  the  reading  on  the  mark  to  get  the  azimuth. 

It  is  necessary,  of  course,  to  have  the  mark  near  enough  to  the  meridian  of  the  instrument 
to  fall  within  the  field  that  can  be  measured  by  means  of  the  reticle  or  with  the  micrometer  wire. 
It  is  best,  in  the  case  of  the  transit  micrometer,  to  place  the  mark  so  nearly  in  the  meridian 
that  its  image  will  fall  inside  the  range  of  the  comb,  so  that  the  number  of  turns  of  the  microme- 
ter screw  may  be  readily  counted  between  the  pointings  in  the  direct  and  reversed  positions. 
The  mark  may  be  placed  either  to  the  north  or  south  and  should,  if  practicable,  be  at  least  a 
mile  from  the  instrument. 

The  method  of  observing  is  as  follows:  Just  before  beginning  time  observations  with  the 
telescope  band  east,  say,  a  number  of  observations  are  taken  on  the  mark;  the  telescope  is 
reversed  to  the  position  band  west,  and  an  equal  number  of  observations  is  made  on  the  mark. 
The  stars  of  the  first  half  set  are  then  observed,  followed  by  observations  on  the  mark.  The 
telescope  is  next  reversed  to  the  position  band  east,  the  mark  observed,  and  then  the  stars  of 
the  second  half  set  are  taken.  Finally,  observations  are  taken  on  the  mark,  the  telescope  is 
reversed  to  position  band  west,  and  the  same  number  of  observations  is  made  on  the  mark. 
This  completes  the  first  set  of  azimuth  observations,  and  the  observations  on  the  stars  for  a  full 
time  set. 

The  mean  of  all  the  readings  on  the  mark  band  east,  is  adopted  as  the  final  value  in  this 
position  of  the  axis  and,  similarly,  the  mean  is  taken  for  all  readings  with  band  west.  The 
mean  of  these  two  adopted  values  for  band  east  and  band  west  gives  the  reading  of  the  colli- 
mation  axis,  and  the  difference  between  either  of  the  two  values  and  the  mean  is  the  angle 
between  the  mark  and  the  collimation  axis  of  the  telescope.  This  angle,  combined  with  the 
azimuth  constant  of  the  time  set,  gives  the  azimuth  of  the  mark.  The  angle  is  observed  as  so 
many  turns  of  the  micrometer  head  or  screw,  or  spaces  of  the  reticle.  This  angle  is  considered 
to  be  positive  when  the  mark  is  east  of  the  colh'mation  axis,  when  pointing  south,  or  west  of 
that  axis  when  pointing  north.  To  this  angle  (reduced  to  seconds  of  time)  is  added  algebraically 
the  azimuth  constant,  a  (see  p.  25),  derived  from  the  computation  of  the  time  set.  This 
azimuth  constant  is  the  angle  between  the  meridian  and  the  collimation  axis.  It  is  considered 
to  be  positive  if  the  collimation  axis  is  east  of  the  meridian,  with  the  telescope  pointing  south, 
or  if  the  axis  is  west  of  the  meridian  with  the  telescope  north. 

If  the  mark  is  much  out  of  the  horizon  of  the  instrument,  readings  of  the  striding  level 
should  be  made  while  observing  on  the  mark,  and  its  elevation  should  be  measured  roughly 
with  the  finder  circle.  The  correction  for  inclination  of  axis  is  applied  as  on  page  145  and  the 
reduction  to  the  horizon,  of  the  angle  between  mark  and  collimation  axis,  is  made  as  on  page  157. 

If  readings  on  the  mark  are  obtained  in  only  one  position  of  the  telescope  axis,  it  will  be 
necessary  to  take  into  consideration  the  collimation  constant  of  the  time  set  and  the  equatorial 
interval 1  of  the  assumed  zero  as  well  as  the  azimuth  constant.  The  reading  on  the  mark  made 
with  the  micrometer  screw,  or  estimated  on  the  reticle,  is  referred  to  some  assumed  zero  of  the 
screw  or  diaphragm.  Combining  the  angle  between  the  mark  and  this  zero  with  the  equatorial 
interval  of  the  zero  gives  the  angle  between  the  mark  and  the  line  of  collimation.  This  latter 
angle,  combined  with  the  collimation  constant  of  the  time  set,  gives  the  angle  between  the 
mark  and  the  collimation  axis.  This  last  angle,  the  angle  between  the  mark  and  the  collimation 
axis,  combined  with  the  azimuth  constant  of  the  time  set,  gives  the  desired  angle  between  the 
mark  and  the  meridian.  That  part  of  the  azimuth  angle  which  lies  between  the  collimation 
axis  of  the  telescope  and  the  mark  must  be  reduced  to  the  horizon  if  the  mark  is  not  in  the 
horizontal  plane  of  the  instrument.  Any  inclination  cf  the  horizontal  axis  must  be  corrected 
for,  as  explained  on  page  145. 

1  This  is  the  angle  between  the  mean  position  of  the  micrometer  wire  or  the  mean  lines  of  the  reticle  and  the  assumed  zero.    See  p.  32. 
8136°— 13 11 


162  U.   S.   COAST  AND  GEODETIC   SURVEY  SPECIAL  PUBLICATION   NO.   14. 

The  following  examples  with  explanations  will  show  this  method  of  determining  azimuth : 
Example  of  record — Readings  on  azimuth  mark. 

TRANSIT  MICROMETER. 

[Station,  Fairbanks,  Alaska.    Date,  Aug.  9, 1910.   Observer,  E.  Smith.   Instrument:  Transit  No.  18,  with  transit  micrometer.   Mark  to  northward.] 


Before  observations  for  time  on 
first  half-set 

Between  the  two  half-sets 

After  observations  for 
time  on  second  half-set 

Band         East 

West 

West 

East 

East 

West 

T 

T 

T 

T 

T 

T 

+5.  050 

+0.  952 

+0.  890 

+5.  050 

+5.  120 

+  1.000 

5.070 

0.915 

0.960 

5.070 

5.090 

0.946 

5.110 

0.940 

0.950 

5.  093 

5.121 

0.985 

5.110 

0.990 

0.  965 

5.082 

5.120 

0.930 

5.040 

0.920 

0.938 

5.060 

5.068 

0.985 

5.020 

0.990 

0.910 

5.049 

5.140 

0.982 

5.  055 

0.930 

0.970 

5.  023 

5.140 

0.960 

5.110 

0.  930 

0.959 

5.100 

5.110 

0.930 

5.  090 

0  950 

0.960 

5.110 

5.080 

0.959 

5.  120 

0.985 

0.958 

5.098 

5.090 

0.967 

Means:  +5.  078 

+0.  947 

+0.  946 

+5.  074 

+5.  108 

+0.  946 

Computation  of  azimuth  from  time  observations. 

TRANSIT  MICROMETER. 

[Fairbanks,  Alaska,  1910.    Transit  No.  18.    Equatorial  interval  of  one  turn  of  micrometer,  2».826.    Mark  to  northward.] 


Date 

August  8 

August  8 

August  9 

Band 

East 

West 

East 

West 

East 

West 

T           s 

T            s 

T            s 

T            s 

T            s 

T           s 

Mean  reading  on  mark 
Mean  reading  of  E.  and  W.  (reading 
of  collimation  axis) 

5.074 
3.048 

1.023 
3.048 

5.  067 
3.032 

0.9% 
3.032 

5.087 
3.016 

0.94B 
3.016 

Angle,  mark  to  collimation  axis 

-2.  026-  -5.  73 

-2.  025=  -5.  72 

-2.035-  —  5.75 

-2.  036=  -5.  75 

-2.  071-  —5.85 

-2.  070-  -5.  85 

a  (from  time  set) 

-0.16 

-0.36 

—0.21 

—0.25 

—0.04 

—0.12 

Angle,  mark  to  meridian 

-5.89 

-6.08 

-5.96 

-6.00 

-5.  89 

-5.97 

Mean  for  set  (in  time) 

-5«.9S 

-5«.98 

-5«.93 

Mean  for  set  (in  arc) 

-89".7 

-89".7 

-89".0 

Mean  azimuth  of  mark  east  of  north,  V  29".5. 
Correction  for  elevation  of  mark,  0.0. 
Reduction  to  mean  position  of  pole,1  +0.8. 
Azimuth  of  mark,  180°  01'  30".3. 

The  comb  should  be  considered  as  being  numbered  from  one  side  to  the  other  and  in  such  a 
way  that  the  numbers  increase  with  increasing  numbers  on  the  micrometer  head  as  the  wire 
is  moved  across  the  field.  For  convenience  the  first  tooth  may  be  given  the  number  1  rather 
than  zero.  The  observer  in  the  field  must  note  in  the  record  for  one  position  of  the  telescope 
(band  west  or  east)  whether  the  line  of  sight  points  farther  east  or  west  with  increasing  readings 
on  the  micrometer  head. 

In  the  example  above,  with  band  east,  the  readings  increase  on  the  micrometer  head  as  the 
line  of  sight  moves  toward  the  east.  That  is,  for  the  reading  of  five  turns,  band  east,  the  line 
of  sight  is  about  two  turns  east  of  the  collimation  axis.  With  band  west  increasing  readings 
correspond  to  a  motion  of  the  line  of  sight  toward  the  west,  a  reading  of  one  turn,  band  west, 
corresponding  to  a  postion  of  the  line  of  sight  of  about  two  turns  east  of  the  collimation  axis. 

A  set  of  azimuth  observations  was  made  with  each  of  two  time  sets  on  August  8. 


i  See  Astronomische  Nachrlchten  No.  4504. 


DETERMINATION   OF   AZIMUTH. 


163 


Computation  of  azimuth  from  time  observations. 

DIAPHRAGM. 
|St.  Michael,  Alaska,  1898.    Meridian  telescope  No.  13.    Equatorial  interval  of  one  space  of  reticle,  3-. 455.    Mark  to  southward.] 


Date 

July  13 

July  14 

July  1.5 

Clamp 

East 

West 

East 

West 

East 

West 

Spaces         s 

Spaces      s 

Spaces       s 

Spaces        s 

Spaces        s 

Spaces       s 

Angle,  mark  to  center  line 

-0.20=  —  0.69 

0.00-    0.00 

-0.175=—  0.60 

-0.025=  -0.09 

-0.  75=  -2.  59 

-0.  15=  -0.  52 

Mean  of  E  and  W 

-0.34 

-0.34 

-0.34 

—0.34 

-1.56 

-1.56 

(  Angle  mark  to  collimation  axis) 

a  (from  time  set) 

+0.39 

+0.86 

+0.40 

+0.72 

+1.75 

+1.63 

Angle,  mark  to  meridian 

+0.05 

+0.52 

+0.06 

+0.38 

+0.19 

+0.07 

Mean  for  set  (in  time) 

+0-.28 

+0-.22 

+0".  13 

Mean  for  set  (in  are) 

+4".  2 

+3".  3 

+2".0 

Date 

July  18 

Sept.  13 

Sept.  17 

Clamp 

East 

West 

East 

West 

.     East 

West 

Spaces        s 

Spaces        t 

Spaces       8 

Spaces       s 

Spaces        s 

Spaces        » 

Angle,  mark  to  center  line 

-0.975-  -3.  37 

-0.05-  -0.17 

0.00=    0.00 

0.00-    0.00 

+0.  25=  +0.  86 

+0.825-  +  2.  85 

Mean  of  E  and  W 

—1.77 

-1.77 

0.00 

0.00 

+1.86 

+1.86 

(Angle  mark  to  collimation  axis) 
a  (from  time  set) 

+2.78 

+2.64 

+0.41 

+0.06 

-2.01 

-1.42 

Angle,  mark  to  rrferidian 

+  1.01 

+0.87 

+0.41 

+0.06 

—0.15 

+0.44 

Mean  for  set  (in  time) 

+0".94 

+0».  24 

+0>.  14 

Mean  for  set  (in  arc) 

+  14".l 

+3".  6 

+2".l 

Final  mean,  mark  east  of  south,       0°  00'  04". 9 
Correction  for  elevation  of  mark  0.0 

Azimuth  of  mark  359°  59'  55".l 

There  is  no  essential  difference  between  the  above  method  and  that  with  the  transit  microm- 
eter. The  angle  between  the  mark  and  the  center  line  of  the  diaphragm  is  estimated  in  spaces 
of  the  reticle.  The  accuracy  of  the  resulting  azimuth  in  this  case  as  well  as  in  that  of  the 
transit  micrometer  depends  largely  on  the  accuracy  with  which  the  azimuth  constant  is  deter- 
mined from  the  time  observations.  The  effect  of  errors  of  pointing  and  reading  on  the  mark 
may  be  made  relatively  small  by  repeated  observations. 

The  work  of  the  Latitude  Service  of  the  International  Geodetic  Association  began  in  1899, 
so  it  is  only  for  observations  made  after  that  year  that  a  satisfactory  reduction  can  now  be  made 
to  the  mean  position  of  the  pole.  It  is  probable  that  in  a  few  years  a  reliable  value  of  this 
reduction  can  be  had,  based  on  theoretical  grounds. 

Computation  of  azimuth  from  time  observations. 

DIAPHRAGM. 

[St.  Michael,  Alaska,  1898.    Meridian  telescope  No.  13.    Readings  on  mark  in  only  one  position  of  telescope  axis.    Equatorial  interval  of  one 

space  of  reticle,  3>.455.    Mark  to  southward.] 


Date 

July  13 

July  14 

Clamp 

East 

East 

Spaces        s 

Spaces         s 

Mark  east  of  center  line 

-0.20=  -0.69 

-0.175=  -0.60 

Eq.  interval  of  center  line 

0.00 

0.00 

c 

+0.12 

+0.18 

a 

+0.39 

+0.40 

Mark  east  of  south 

-0.18 

-0.02 

Mark  east  of  south 

-2".  7 

-0".  3 

164  U.   S.   COAST  AND  GEODETIC   SURVEY  SPECIAL  PUBLICATION   NO.   14. 

The  above  is  taken  from  the  example  already  given  for  observations  in  both  positions  of  the 
telescope.  In  this  case  of  deriving  the  azimuth  from  observations  on  the  mark  in  only  one 
position  of  the  axis,  the  equatorial  interval  of  the  assumed  zero  and  the  collimation  constant  of 
the  time  set  must  be  applied  to  the  reading  on  the  mark.  The  collimation  constant  is  applied 
with  the  same  sign  as  derived  from  the  computation  of  the  time  set  when  the  observations  on 
the  mark  are  made  with  band  west,  mark  south,  and  with  the  opposite  sign  when  made  witli 
band  east,  mark  south.  The  equatorial  interval,  i,  of  the  assumed  zero  of  the  reticle  or  microm- 
eter is  considered  positive  when  west  of  the  mean  line  or  position,  band  west.  It  follows,  then, 
that  when  i  and  c  are  combined  in  the  azimuth  angle  they  are  applied  with  opposite  signs. 
Defining  the  measured  angle  between  the  mark  and  the  assumed  zero  as  positive  when  the  mark 
is  east  of  the  zero,  pointing  south,  and  using  a,  c,  and  i,  with  their  conventional  signs,  the  follow- 
ing general  expressions  cover  all  cases  : 


M    ,  j  .  .  .      =          -  {aw+  (M  +  c-i)  sec  h}l5 

JBandE    .  .  .  «  =  360°-  {a,  +  (M-e+i)  sec 


Mark,,orthBandW  '  '  '  "=  180°-  {aw+  (Jf-c  +  i)  sec 

^JBandE    .  .  .  «=180°-  K  +  (M+c-i)  sec  A}  15 

aw  and  aE  are  the  azimuth  constants  from  the  time  set.  M  is  the  angle  (in  seconds  of  time) 
between  the  mark  and  the  assumed  zero  of  the  micrometer  or  diaphragm.  It  is  assumed  to 
be  positive  when  the  mark  is  east  of  the  zero  when  pointing  south.  It  is  also  positive  when 
the  mark  is  west,  pointing  north,  c  is  the  collimation  constant  of  the  time  set.  i  is  the  equato- 
rial interval,  in  seconds  of  tune,  between  the  mean  position  of  the  micrometer  wire  and  the 
assumed  zero  of  the  micrometer,  or  between  the  mean  line  of  the  reticle  and  the  assumed  zero. 
h  is  the  angle  of  elevation  or  depression  of  the  mark.  The  quantity  to  be  subtracted  from  360° 
or  180°  is  in  seconds  of  arc. 

CORRECTION  FOR  ELEVATION  OF  MARK. 

When  the  object  used  as  an  azimuth  mark  is  at  a  considerable  elevation,  it  is  necessary  to 
apply  a  correction  to  obtain  the  astronomic  azimuth  of  the  projection  of  the  mark  on  the  sphe- 
roidal surface  of  reference.  This  correction,  in  seconds,  is: 


in  which  e2  is  the  square  of  the  eccentricity  and  a  the  semi-major  axis  of  the  spheroid  of  refer- 
ence; <j>  is  the  latitude  of  the  observing  station;  a  is  the  azimuth  of  the  line  to  the  mark;  and 
h  is  the  elevation  of  the  mark.  For  h  in  meters,  and  Clarke's  1866  dimensions  of  the  spheroid, 
as  stated  in  meters,  this  expression  becomes: 

+  0'^.000109  h  cos2  0  sin  la,  or 
+  [  6.0392]  h  cos2  <£  sin  2a, 

where  the  number  in  brackets  is  a  logarithm,  the  dash  over  the  characteristic  indicating  that 
10  is  to  be  substracted  from  it.  The  sign  of  the  expression  shows  that  when  the  mark  is  either 
southwest  or  northeast  of  the  observing  station  the  observed  azimuth  of  the  mark  must  be 
increased  to  obtain  the  correct  azimuth,  while  for  mark  northwest  or  southeast,  the  observed 
azimuth  must  be  decreased. 

CORRECTION  FOR  VARIATION  OF  THE  POLE. 

A  correction  is  necessary  to  reduce  the  observed  astronomic  azimuth  to  the  mean  position 
of  the  pole.  This  correction  may  amount  to  a  half-second  or  more  for  points  in  the  northern 
part  of  the  United  States.  The  secant  of  the  latiude  is  a  factor  of  the  correction,  so  the  value 
becomes  larger  for  the  higher  latitudes.  (See  p.  85.) 


DETERMINATION   OF   AZIMUTH. 


165 


Log 


1-a 


Log  a 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Proportional  parts 

9.00 

0.045758 

5869 

5980 

6092 

6204 

6317 

6429 

6542 

6656 

6769 

111 

108 

105 

102  |   99 

8.99 

0.044660 

4769 

4878 

4987 

5096 

5205 

5315 

5425 

5536 

5647 

1 

11.1 

10.8 

10.5 

10  2 

9.9 

98 

3591 

3697 

3803 

3909 

4016 

4122 

4229 

4337 

4444 

4552 

2 

22.2 

21.6 

21.0 

20.4 

19.8 

97 

2549 

2652 

2755 

2858 

2962 

3066 

3171 

3275 

3380 

34S6 

3 

33.3 

32.4 

31.5 

30.6 

29.7 

96 

1532 

1633 

1733 

1834 

1936 

2037 

2139 

2241 

2343 

2446 

4 

44.  4 

43.2 

42.0 

40  g 

39  6 

95 

0.040541 

0639 

0737 

0836 

0935 

1034 

1133 

1232 

1332 

1432 

5 

55.5 

54.0 

52.5 

51.0 

49.5 

94 

0.  039575 

9670 

9766 

9862 

9959 

0055 

8152 

0249 

8346 

0443 

6 

7 

66.6 

77.7 

64.8 
75  6 

63.0 
73.5 

61.2 
71.4 

59.4 
69.3 

93 

8633 

8726 

8819 

8913 

9007 

9101 

9195 

9290 

9385 

9480 

8 

88.8 

86.4 

84.0 

81.6 

79.2 

92 

7714 

7805 

78% 

7987 

8079 

8171 

8263 

8355 

8447 

8540 

9 

99.9 

97.2 

94.5 

91.8 

89.1 

91 

6818 

6907 

6996 

7085 

7174 

7263 

7353 

7443 

7533 

7624 

8.90 

0.  035944 

6031 

6118 

6204 

6291 

6379 

6466 

6554 

6642 

6730 

96 

93 

90 

87 

84 

89 

5092 

5177 

5261 

5346 

5431 

5516 

5601 

5687 

5772 

5858 

88 
87 
86 
85 

4261 
3451 
2660 
0.031888 

4343 
3531 

2738 
1965 

4426 
3611 

2816 
2041 

4508 
3692 
2896 
2118 

4591 
3772 
2974 
2195 

4674 
3853 
3053 
2272 

4757 
3934 
3132 
2349 

4841 
4016 
3211 
2426 

4924 
4097 
3291 
2504 

5008 
4179 
3371 
2582 

1 
2 
3 
4 

5 

9.  6 
19.2 
28.8 
38.4 
48.0 

9.  A 
18.6 
27.9 
37.2 
46.5 

9.  0 
18.0 
27.0 
36.0 
45.0 

8.  7 
17.4 
26.1 
34.8 
43.5 

g.  4 
16.8 
25.2 
33.6 
42.0 

84 
83 
82 
81 

1136 
0402 
0.029685 
8987 

1210 

0474 
9756 
9056 

1285 
0547 
9827 
9125 

1360 
0620 
9898 
9194 

1435 
0693 
9970 
9264 

1510 
0766 
0041 
9334 

1585 
0840 
0113 
9404 

1660 
0914 
0185 
9474 

1736 
0987 
0257 
9544 

1812 
1061 
0329 

%i5 

6 

7 
g 
9 

57.6 
67.2 
76.8 
86.4 

55.8 
65.1 
74.4 
83.7 

54.0 
63.0 
72.0 
81.0 

52.2 
60.9 
69.6 
78.3 

50.4 
58.8 
67.2 
75.6 

8.80 

0.028305 

8372 

8440 

8508 

8576 

8644 

8712 

8780 

8849 

8918 

81 

78 

75 

72 

69 

79 

78 

7640 
6990 

7705 
7055 

7771 
7119 

7838 
7183 

7904 
7248 

7970 
7313 

8037 
7378 

8103 
7443 

8170 
7509 

8237 
7574 

1 

8.1 

7.8 

7.5 

7.2 

6.9 

77 

6357 

6420 

6482 

6545 

6608 

6672 

6735 

6799 

6862 

(i'J-'li 

2 

16.2 

15.6 

15.0 

14.4 

13.8 

76 

5739 

5800 

5861 

r,\m 

5984 

6046 

6108 

6170 

6232 

(12114 

3 

24.3 

23.4 

22.5 

21.6 

20.7 

75 

0.  025136 

5195 

5255 

5315 

5375 

5435 

5496 

5556 

5617 

5678 

4 

32.4 

31.2 

30.0 

28.8 

27.6 

5 

40.5 

39.0 

37.5 

36.0 

34.5 

74 

4547 

4605 

4664 

4722 

4781 

4840 

4899 

4958 

5017 

5076 

6 

48.6 

46.8 

45.0 

43.2 

41.4 

73 

3973 

4029 

4086 

4143 

4201 

4258 

4316 

4373 

4431 

4489 

7 

56.7 

54.6 

52.5 

50.4 

48.3 

72 

3412 

3467 

3523 

3579 

3635 

3691 

3747 

3803 

3859 

3916 

g 

64.8 

62.4 

60.0 

57.6 

55.2 

71 

2865 

2919 

2973 

3027 

3082 

3137 

3191 

3246 

3301 

3357 

9 

72.9 

70.2 

67.5 

64.8 

62.1 

8.70 

0.022331 

2383 

2436 

2489 

2543 

2596 

2649 

2703 

2757 

2811 

fiO 

1  '11'  A 

OfwjQ 

9191 

91  7Q 

99OK 

tyvjo 

H 

63 

60 

57 

55 

Otf 

68 

1809 
1301 

1861 
1351 

1913 
1401 

liW>4 

1452 

2016 
1503 

^UDo 

1553 

xm 

1604 

•Xfa 

1655 

•BQ 

1707 

££lO 

1758 

1 

6.6 

6.3 

6.0 

5.7 

5.5 

67 

0804 

0853 

0902 

0952 

1001 

1051 

1100 

1150 

1200 

1250 

2 

13.2 

12.6 

12.0 

11.4 

11.0 

66 

0319 

0367 

0415 

0463 

0512 

0560 

0609 

0657 

0706 

0755 

3 

19.8 

18.9 

18.0 

17.1 

16.5 

65 

0.019846 

9893 

9940 

9987 

0034 

0081 

0128 

6176 

0223 

0271 

4 

26.4 

25.2 

24.0 

22.8 

22.0 

5 

33.0 

31.5 

30.0 

28.5 

27.5 

64 

9384 

9430 

9475 

9521 

9567 

9613 

9660 

9706 

9752 

9799 

6 

39.6 

37.8 

36.0 

34.2 

33.0 

63 

8933 

8978 

9022 

9067 

9112 

9157 

9202 

9247 

9293 

9338 

7 

46.2 

44.1 

42.0 

39.9 

38.5 

62 

8493 

8536 

85SO 

8S24 

8667 

8711 

8755 

8800 

8844 

8888 

8 

52.8 

50.4 

48.0 

45.6 

44.0 

61 

8063 

8105 

8148 

8191 

8233 

8276 

8319 

8363 

8406 

8449 

9 

59.4 

56.7 

54.0 

51.3 

49.5 

8.60 

0.017643 

7685 

7726 

7768 

7810 

7852 

7894 

7936 

7978 

8020 

53 

51 

49 

47 

45 

59 

7233 

7274 

7315 

7355 

7396 

7437 

7478 

7519 

7560 

7602 

58 

6833 

6873 

6913 

6952 

6992 

7032 

7072 

7112 

7153 

7193 

1 

5.3 

S.I 

4.9 

4.7 

4.5 

57 

6443 

6482 

6520 

6559 

6598 

6637 

6676 

6715 

6755 

6794 

2 

10.6 

10  2 

9  8 

9  4 

9.0 

56 

6062 

6099 

6137 

6175 

6213 

6251 

li2S!) 

6328 

6366 

6404 

3 

15.9 

15.3 

14.7 

14.1 

13.5 

55 

0.015689 

5726 

5763 

5800 

5837 

5874 

5912 

5949 

5986 

6024 

4 

21.2 

20.4 

19.6 

18.8 

18.0 

5 

26.5 

25.5 

24.5 

23  5 

22  5 

54 

5326 

5362 

5398 

5434 

5470 

5507 

5543 

5579 

5616 

5653 

6 

31.8 

30.6 

29.4 

28.2 

27.0 

53 

4971 

5006 

5041 

5077 

5112 

5147 

5183 

5218 

5254 

5290 

7 

37.1 

35.7 

34.3 

32.9 

31.5 

52 

4624 

4659 

4693 

4727 

4762 

4797 

4831 

4866 

4901 

4936 

g 

42.4 

40.8 

39.2 

37.6 

36  0 

51 

4286 

4319 

4353 

4387 

4420 

4454 

4488 

4522 

4556 

4590 

9 

47.7 

45.9 

44.1 

42.3 

40.5 

8.50 

0.013955 

3988 

4021 

4054 

4087 

4120 

4153 

4186 

4219 

4253 

43 

41 

39 

37 

35 

1 

4.3 

4.1 

3.9 

3.7 

3.5 

2 

8.6 

8.2 

7.8 

7.4 

7.0 

3 

12.9 

12.3 

11.7 

11.1 

10.5 

4 

17.2 

16.4 

15.6 

14.8 

14.0 

5 

21.5 

20.5 

19.5 

18.5 

17.5 

6 

25.8 

24.6 

23.4 

22.2 

21.0 

7 

30.1 

28.7 

27.3 

25.9 

24  5 

g 

34.4 

32.  g 

31.2 

21.6 

28.0 

9 

38.7 

36.9 

35.1 

33.3 

31.5 

166 


U.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.   14. 

Loq  -j • 

9  I  —  a 


Log  a 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Proportional  parts 

8.50 

0.013955 

3988 

4021 

4054 

4087 

4120 

4153 

4186 

4219 

4253 

34 

33 

32 

31 

30 

49 

3633 

3665 

3697 

3729 

3761 

3793 

3825 

3858 

3890 

3923 

1 

3.  4 

3.  3 

3.2 

3.  1 

3.0 

48 

3318 

3349  3380 

3411 

3443 

3474 

3506 

3537 

3569 

3601 

2 

6.8 

6.6 

6.4 

6.2 

6.0 

47 

3010 

3040  3071 

3101 

3132 

3163 

3194 

3225 

3256 

3287 

3 

10.  2 

9.  9 

9.6 

9.  3 

9.  0 

r    46 

2709 

2739  2769 

2799 

2829 

2S59 

2889 

2919 

2949 

2979 

4 

13.6 

13^2 

12  8 

12.4 

12.  0 

45 

0.  012416 

2445 

2474 

2503 

2532 

2562 

2591 

2621 

2650 

2680 

5 

17.0 

16.5 

16.0 

15.5 

15.0 

6 

20.4 

19.  8 

19.2 

18.6 

18.0 

44 

2129 

2158 

2186 

2215 

2243 

2272 

2300 

2329 

2358 

2387 

7 

23.8 

23.  1 

22.4 

21.  7 

21.  0 

43 

1849 

1877 

1905 

1933 

1961 

1989 

2017 

2045 

2073 

2101 

8 

27.2 

26  4 

25  6 

24.8 

24.0 

42 

41 

1576 
1309 

1603 
1335 

1630 
1362 

1657 
1388 

1685 
1415 

1712 
1442 

1739 
1468 

1767 
1495 

1794 
1522 

1822 
1549 

9 

30.6 

29.7 

28.8 

27^9 

27!  o 

8.40 

0.011048 

1074 

1100 

1126 

1152 

1178 

1204 

1230 

1256 

1283 

29 

28 

27 

26 

25 

39 
38 

0794 
0545 

0819 
0570 

0844 
0594 

0869 
0619 

0895 
0644 

0920 
0669 

0946 
0694 

0971 
0718 

0997 
0743 

1023 

0769 

1 

2.9 

2.8 

2.7 

2.6 

2.5 

37 

0302 

0326 

0350 

0374 

0399 

0423 

0447 

0472 

0496 

0520 

2 

5.8 

5.6 

5.4 

5.2 

5.0 

36 
35 

0065 
0.009833 

0088 
9856 

0112 
9879 

0135 
9902 

0159 
9925 

0183 
9948 

0207 
9972 

0230 
9995 

0254 
0018 

(127s 
0041 

3 
4 
5 

8.7 
11.6 
14.5 

8.4 
11.2 
14.0 

g.  1 

10.8 
13.5 

7.8 
10.4 
13.0 

7.5 
10.0 
12.5 

34 
33 
32 
31 

9607 
9386 
9170 
8959 

9629 
9408 
9191 
8980 

9652 
9430 
9213 
9001 

9674 
9452 
9234 
9022 

9697 
9474 
9256 
9043 

9719 
9496 
9277 
9064 

9742 
9518 
9299 
9085 

9765 
9540 
9320 
9106 

9787 
9562 
9342 
9127 

9810 
9584 
9364 
9149 

g 

17.4 
20.3 
23.2 
26.1 

16.8 
19.6 
22.4 
25.2 

16.2 
18.9 
21.6 
24.3 

15.6 
18.2 
20.8 
23.4 

15.0 
17.5 
20.0 
22.5 

8.30 

0.  008753 

8773 

8794 

8814 

8835 

8855 

8876 

8897 

8917 

8938 

24 

23 

22 

21 

20 

29 
28 

8552 
8355 

8572 
8375 

8592 
8394 

8612 
8414 

8632 
8433 

8652 
8453 

8672 
8473 

8692 
8492 

8712 
8512 

8733 

s.vu 

1 

2.4 

2.3 

2.2 

2.1 

2.0 

27 
26 

8163 
7976 

8182 
7994 

8201 
8013 

8220 
8031 

8050 

8259 
8069 

8278 

MISS 

8297 
8106 

8316 
8125 

8336 
8144 

3 

7.2 

6.9 

6.6 

6.3 

6.0 

25 

0.  007792 

7811 

7829 

7847 

7865 

7884 

7902 

7920 

7939 

7957 

4 

9.6 

9.2 

8.8 

8.4 

8.0 

5 

12.0 

11.5 

11.0 

10.5 

10.0 

24 

7614 

7631 

7649 

7667 

7685 

7702 

7720 

7738 

7756 

7774 

6 

14.4 

13.8 

13.2 

12.6 

12.0 

23 

7439 

7456 

7473 

7491 

7508 

7526 

7543 

7561 

7578 

7596 

7 

16.8 

16.1 

15.4 

14.7 

14.0 

22 

7268 

7285 

7302 

7319 

7336 

7353 

7370 

7387 

7404 

7421 

g 

19.2 

18.4 

17.6 

16.8 

16.0 

21 

7101 

7118 

7134 

7151 

7167 

7184 

7201 

7218 

7234 

7251 

9 

21.6 

20.7 

19.8 

18.9 

18.0 

8.20 

0.  006938 

6954 

6971 

6987 

7003 

7019 

7036 

7052 

7068 

7085 

COCO 

19 

18 

17 

16 

15 

19 

18 

6779 
6624 

6639 

6811 
6654 

6670 

6685 

oooo 
6701 

6716 

6890 

6732 

6748 

6763 

1 

1.9 

1.8 

1.7 

1.6 

1.5 

17 

6472 

6487 

Coon 

6502 

fl-lSO 

6517 
6367 

6532 

6547 

6562 

6578 

6593 

6608 

2 

3.8 

3.6 

3.4 

3.2 

3.0 

16 

15 

6323 
0.006178 

UJOO 

6193 

Do  0,5 

6207 

6221 

6236 

6250 

6265 

6279 

6442 
6294 

6457 
6309 

4 

7.6 

7.2 

6.8 

6.4 

6.0 

5 

9.5 

9.0 

8.5 

8.0 

7.5 

14 

6037 

6051 

6065 

6079 

6093 

6107 

6121 

6135 

6150 

6164 

6 

11.4 

10.8 

10.2 

9.6 

9.0 

13 

5898 

5912 

5926 

5940 

5353 

5967 

5981 

5995 

6009 

6023 

7 

13.3 

12.6 

11.9 

11.2 

10.5 

12 

5763 

5777 

5790 

5803 

5817 

5830 

5844 

5857 

5871 

5885 

8  15.2 

14.4 

13.6 

12.8 

12.0 

11 

5631 

5644 

5657 

5670 

5684 

5697 

5710 

5723 

5737 

5750 

9  17.1 

16.2 

15.3 

14.4 

13.5 

8.10 

0.  005502 

5515 

5528 

5541 

5553 

5566 

5579 

5592 

5605 

5618 

14 

13 

12 

11 

10 

09 
08 

5376 
5253 

5389 
5265 

5401 
5277 

5414 
5290 

5426 
5302 

5439 
5314 

5451 
5327 

5464 
5339 

5477 
5351 

5489 
5364 

1 

1.4 

1.3 

1.2 

1.1 

1.0 

07 

5133 

5145 

5157 

5169 

5181 

5193 

5205 

5217 

521*1 

5241 

2 

2.8 

2.6 

2.4 

2.2 

2.0 

06 

5015  5027 

5038 

5050 

5062 

5074 

5085 

5097 

5109 

5121 

3 

4.2 

3.9 

3.6 

3.3 

3.0 

05 

4900  4912 

4923 

4935 

4946 

4957 

4969 

4980 

4992 

5004 

4 

5.6 

5.2 

4.8 

4.4    4.0 

5 

7.0  '  6.5   6.0 

5.  5    5.  0 

04 

4788  I  4799 

4810 

4822 

4833 

4844 

4855 

4866 

4K7S 

4889 

6 

8.4   7.8   7.2 

6.  6    6.  0 

03 

4679 

4690  4700 

4711 

4722 

4733 

4744 

4755 

4766 

4777 

7 

9.8 

9.  1  i  8.  4 

7.  7    7.  0 

02 

4572 

4582  4593 

4603 

4614 

4625 

4636 

4646 

4657 

4668 

g 

11.2 

10.4   9.6   8.8 

8.0 

01 

4467 

4477  4488 

4498 

4509 

4519 

4529 

4540 

4550 

4561 

9 

12.6 

11.7  10.8  ,  9.9 

9.0 

8.00 

0.004365 

4375  4385 

4395 

4405 

4416 

4426 

4436 

4446 

4457 

DETEKMINATION   OF  AZIMUTH. 
1 


167 


Log 


I—a 


Log  a 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Proportional  parts 

8.00 

0.004365 

4375 

4385 

4395 

4405 

4416 

4426 

4436 

4446 

4457 

7.99 

4265 

4275 

4285 

4295 

4305 

4315 

4325 

4335 

4345 

4355 

11 

10 

98 

4167 

4177 

4187 

4196 

4206 

4216 

4226 

4235 

4245 

4255 

97 

4072 

4082 

4091 

4100 

4110 

4119 

4129 

4139 

4148 

4158 

°~^^ 

96 

3979 

3988 

3997 

4007 

4016 

4025 

4035 

4044 

4053 

4063 

1 

1.1 

1.0 

95 

0.003888 

3897 

3906 

3915 

3924 

3933 

3942 

3951 

3961 

3970 

2 

2.2 

2.0 

3 

3.3 

3.0 

94 

3799 

3808 

3817 

3826 

3834 

3843 

3852 

3861 

3870 

3879 

4 

4.4 

4.0 

93 

3712 

3721 

3729 

3738 

3747 

3755 

3764 

3773 

3782 

3790 

5 

5.5 

5.0 

92 

3627 

3636 

3644 

3653 

3661 

3670 

3678 

3687 

3695  3704 

6 

6.6 

6.0 

91 

3545 

3553 

3561 

3569 

3577 

3586 

3594 

3602 

3611  1  3619 

7 

7.  7 

7  0 

7.90 

0.003463 

3472 

3480 

3488 

3496 

3504 

3512 

3520 

3528  3536 

8 

8.8 

8.0 

9 

9.9 

9.0 

89 

3384 

3392 

3400 

3408 

341(1 

3424 

3432 

3440 

3448  3456 

88 

3307 

3315 

3322 

3330 

333S 

3345 

3353 

3361 

3369  :  3377 

87 

3231 

3239 

3246 

3254 

3261 

3269 

3277 

3284 

3292 

3299 

86 

3158 

3165 

3172 

3180 

3187 

3194 

3202 

3209 

3217 

3224 

g 

8 

85 

0.003086 

3093 

3100 

3107 

3114 

3121 

3129 

3136 

3143 

3150 

84 

3015 

3022 

3029 

3036 

3043 

3050 

3057 

3064 

3071  !  3078 

0  9 

0  8 

83 

2946 

2953 

20(10 

2967 

2974 

2980 

2987 

2994 

3001 

3008 

2 

1.  8 

1.  6 

82 

2879 

2886 

2892 

2899 

2906 

2912 

2919 

2926 

2933 

2939 

27 

24 

81 

2813 

2820 

2826 

2833 

2839 

2X46 

2852 

2859 

2xia> 

2872 

.  / 

•  1 

39 

7.80 

0.002749 

2755 

2762 

27<>.x 

2774 

2781 

2787 

2794 

2800 

2807 

4 
5 

3.  6 

4.5 

.  t> 

4.0 

79 

2686 

2692 

2699 

2705 

2711 

2717 

2724 

2730 

2736 

2743 

6 

5.4 

60 

4.8 
5  6 

78 

2625 

2631 

2637 

2643 

2649 

2655 

2661 

2668 

2674 

2680 

i 

.  o 

79 

6  A 

77 

2565 

2571 

2577 

2583 

2589 

2595 

2601 

2607 

2613 

2619 

o 

.  ^ 
81 

•  1 

7  9 

76 

2506 

2512 

2518 

2524 

2530 

2535 

2541 

2547 

2553 

2559 

.  1 

1.  & 

75 

0.002449 

2455 

2460 

2466 

2472 

2478 

2483 

2489 

2495 

2501 

74 

2393 

2399 

2404 

2410 

2415 

2421 

2427 

2432 

2438 

2443 

7 

C 

73 

2339 

2344 

2349 

2355 

2360 

2366 

2371 

2377 

2382 

2388 

72 

2285 

2290 

2296 

2301 

2306 

2312 

2317 

2322 

2328 

2  33 

71 

2233 

2238 

2243 

2249 

2254 

2259 

2264 

2269 

2275 

22X0 

7.70 

0.002182 

2187 

2192 

2197 

2202 

2207 

2213 

2218 

2223 

222S 

1 
2 

0.7 
1.4 

0.6 
1.2 

69 

2132 

2137 

2142 

2147 

2152 

2157 

2162 

2167 

2172 

2177 

3 

2.1 

1.8 

68 

2084 

20XX 

2093 

20!  IS 

2103 

2108 

2113 

2118 

2122 

2127 

4 

2.8 

2.4 

67 

2036 

2041 

2046 

2050 

2055 

2060 

2085 

2069 

2074 

2079 

5 

3.5 

3.0 

66 

1990 

1994 

1999 

2003 

2008 

2013 

2017 

2022 

2027 

2031 

6 

4.2 

3.6 

65 

0.  001944 

1949 

1953 

1958 

1962 

1967 

1971 

1976 

1980 

1985 

7 
8 

4.9 
5.6 

4.2 

4.8 

64 

1900 

1904 

1909 

1913 

1918 

1922 

1926 

1931 

1935 

1940 

9 

6.3 

5.4 

63 

1857 

1861 

1865 

1869 

1874 

1878 

1XX2 

1887 

1891 

1896 

62 

1814 

1818 

1823 

1827 

1X31 

1835 

1840 

1844 

1848 

1852 

61 

1773 

1777 

1781 

1785 

1789 

1793 

1798 

1802 

1806 

1810 

7.60 

0.  001732 

1736 

1740 

1744 

1748 

1753 

1757 

1761 

1765 

1769 

5 

4 

59 

1693 

1697 

1701 

1705 

1709 

1713 

1716 

1720 

1724 

1728 

58 

1654 

1658 

1662 

1666 

1670 

1673 

1677 

1681 

1685 

1689 

1 

0.5 

0.4 

57 

1617 

1620 

1624 

1628 

1632 

1635 

1639 

1643 

1647 

1650 

2 

1.0 

0.8 

56 

1580 

1583 

1587 

1591 

1594 

1598 

1602 

1605 

1609 

1613 

3 

1.5 

1.2 

55 

0.001544 

1547 

1551 

1554 

1558 

1562 

1565 

1569 

1572 

1576 

4 

2.0 

1.6 

5 

2.5 

2.0 

54 

1508 

1512 

1515 

1519 

1522 

1526 

1529 

1533 

1537 

1540 

6 

3.0 

2.4 

53 

1474 

1477 

1481 

1484 

1488 

1491 

1495 

1498 

1502 

1505 

7 

3.5 

2.8 

52 

1440 

1444 

1447 

1450 

1454 

1457 

1461 

1464 

1467 

1471 

8 

4.0 

3.2 

51 

1408 

1411 

1414 

1417 

1421 

1424 

1427 

1431 

1434 

1437 

9 

4.5 

3.6 

7.50 

0.001376 

1379 

1382 

1385 

1388 

1391 

1395 

1398 

1401 

1404 

168 


U.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.    14. 


Log  7 

*  1 — a 


Logo 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Proportional  parts 

7.50 

0.  001376 

1379 

1382 

1385 

1388 

1391 

1395 

1398 

1401 

1404 

49 

1344 

1347 

1350 

1354 

1357 

1360 

1363 

1366 

1369 

1372 

48 

1314 

1317 

1320 

1323 

1326 

1329 

1332 

1335 

1338 

1341 

47 

1284 

1287 

1290 

1292 

1295 

1298 

1301 

1304 

1307 

1311 

46 

1254 

1257 

1260 

1263 

1266 

1269 

1272 

1275 

1278 

1281 

45 

0.001226 

1229 

1231 

1234 

1237 

1240 

1243 

1246 

1249 

1251 

44 

1198 

1201 

1203 

1206 

1209 

1212 

1214 

1217 

1220 

1223 

43 

1170 

1173 

1176 

1179 

1181 

1184 

1187 

1190 

1192 

1195 

42 

1144 

1146 

1149 

1152 

1154 

1157 

1160 

1162 

1165 

1168 

41 

1118 

1120 

1123 

1126 

1128 

1131 

1133 

1136 

1139 

1141 

7.40 

0.  001092 

1095 

1097 

1100 

1102 

1105 

1107 

1110 

1113 

1115 

39 

1067 

1070 

1072 

1075 

1077 

1080 

1082 

1085 

1087 

1090 

38 

1043 

1045 

1048 

1050 

1053 

1055 

1058 

1060 

1062 

1065 

37 

1019 

1022 

1024 

1026 

1029 

1031 

1033 

1036 

1038 

1011 

4 

3 

36 

0.000996 

998 

1001 

1003 

1005 

1008 

1010 

1012 

1015 

1017 

35 
34 

0.000973 
951 

976 
953 

978 
956 

980 
958 

982 
960 

985 
962 

987 
964 

989 
967 

991 
969 

994 
971 

1 

2 

0.4 
0.  8 

0.3 
0.6 

33 

929 

932 

934 

936 

938 

940 

942 

945 

947 

949 

3 

1.2 

0.9 

32 

908 

910 

913 

915 

917 

919 

921 

923 

925 

927 

4 

1.6 

1.2 

31 

888 

890 

892 

894 

896 

898 

900 

902 

904 

906 

5 

2  0 

1.5 

7.30 

0.000867 

869 

871 

873 

875 

877 

879 

882 

884 

886 

6 

2.4 

1.8 

7 

2.8 

2.  1 

29 

848 

850 

852 

854 

855 

857 

859 

861 

863 

865 

8 

3~2 

2.4 

28 

828 

830 

832 

834 

836 

838 

840 

842 

844 

846 

9 

3.6 

2.7 

27 

809 

811 

813 

815 

817 

819 

821 

823 

825 

826 

26 

791 

793 

795 

796 

798 

800 

802 

804 

806 

SOS 

25 

0.  000773 

775 

777 

778 

780 

782 

784 

786 

787 

789 

24 

755 

757 

759 

761 

762 

764 

766 

768 

769 

771 

23 

738 

740 

742 

743 

745 

747 

748 

750 

752 

754 

22 

721 

723 

725 

726 

728 

730 

731 

733 

735 

736 

21 

705 

707 

708 

710 

711 

713 

715 

716 

718 

720 

7.20 

0.000689 

690 

692 

694 

695 

697 

698 

700 

702 

703 

2     1 

19 

673 

675 

676 

678 

fiTQ 

fiS1 

COO 

fifli 

fiftfi 

fiS7 

I 

18 

658 

659 

661 

662 

o/y 

664 

001 
665 

Ooo 

667 

05* 
669 

oso 
670 

DO/ 

672 

17 

643 

644 

646 

647 

649 

650 

652 

653 

655 

656 

1 

0.  2 

0.  1 

16 

628 

630 

631 

633 

634 

635 

637 

638 

640 

641 

2 

0.  4 

00 

0.  2 

0"! 

15 

0.000614 

615 

617 

618 

620 

821 

622 

624 

625 

627 

4 

.  O 

0.8 

•  li 

0.4 

14 

600 

601 

603 

604 

605 

607 

608 

610 

611 

612 

5 

1.0 

0.5 

13 

586 

588 

589 

590 

592 

593 

594 

596 

597 

599 

6 

1.  2 

0.  6 

12 

573 

574 

576 

577 

578 

580 

581 

582 

5S4 

585 

7 

1.  4 

0.  7 

11 

560 

561 

562 

564 

565 

566 

568 

569 

570 

572 

8 

1.  6 

0.  8 

7.10 

0.  000547 

548 

550 

551 

552 

553 

555 

556 

557 

559 

9 

1.8 

0.9 

09 

535 

536 

537 

538 

540 

541 

542 

543 

545 

546 

08 

522 

524 

525 

526 

527 

529 

530 

531 

532 

533 

07 

511 

512 

513 

514 

515 

516 

518 

519 

520 

521 

06 

499 

500 

501 

502 

504 

505 

506 

507 

508 

509 

05 

0.000488 

489 

490 

491 

492 

493 

494 

495 

497 

498 

04 

476 

478 

479 

480 

481 

482 

483 

484 

485 

486 

03 

466 

467 

468 

469 

470 

471 

472 

473 

474 

475 

02 

455 

456 

457 

458 

459 

460 

461 

462 

463 

465 

01 

445 

446 

447 

448 

449 

450 

451 

452 

453 

454 

7.00 

0.000435 

436 

437 

438 

439 

440 

441 

442 

443 

444 

DETERMINATION    OF   AZIMUTH. 
1 


169 


Log  a 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Proportional  parts 

7.00 

0.000435 

436 

437 

438 

439 

440 

441 

442 

443 

444 

• 

10 

9 

6.9 

345 

353 

361 

370 

378 

387 

396 

405 

415 

425 

1 

1.0 

0.9 

8 

274 

280 

287 

294 

301 

308 

315 

322 

330 

337 

2 

2.0 

1.8 

7 

218 

223 

228 

233 

239 

244 

250 

256 

262 

268 

3 

3.0 

2.7 

6 

173 

177 

181 

185 

190 

194 

199 

203 

208 

213 

4 

4.0 

3.6 

5 

0.000137 

141 

144 

147 

151 

154 

158 

161 

165 

169 

5 

5.0 

4.5 

6 

6.0 

5.4 

4 

109 

112 

114 

117 

120 

122 

125 

128 

131 

134 

7 

7.0 

6.3 

3 

87 

89 

91 

93 

95 

97 

100 

102 

104 

107 

8 

8.0 

7.2 

2 

69 

70 

72 

74 

75 

77 

79 

81 

83 

85 

9 

9.0 

8.1 

1 

55 

56 

57 

59 

60 

61 

63 

64 

66 

67 

6.0 

0.000043 

44 

45 

47 

48 

49 

50 

51 

52 

53 

8 

7 

5.9 

34 

35 

36 

37 

38 

39 

40 

41 

41 

42 

~ 

0.  8 

0.  7 

8 

27 

28 

29 

29 

30 

31 

31 

32 

33 

34 

2 

L6 

L4 

7 
6 

22 

17 

22 

18 

23 

18 

23 

19 

24 
19 

24 
19 

25 
20 

26 
20 

26 

21 

27 
21 

3 

A 

2.t 

3  2 

2.1 

2  8 

5 

0.000014 

14 

14 

15 

15 

15 

16 

16 

17 

17 

5 

3!  5 

4 

11 

11 

11 

12 

12 

12 

13 

13 

13 

13 

6 

7 

4^8 
5  6 

4.2 
4  9 

3 

9 

9 

9 

9 

10 

10 

10 

10 

10 

11 

g 

6*4 

5.6 

2 

7 

7 

7 

7 

8 

8 

8 

8 

8 

8 

<j 

7  2 

6  3 

1 

5 

6 

6 

6 

6 

6 

6 

6 

7 

7 

5.0 

0.000004 

4 

5 

5 

5 

5 

5 

5 

5 

5 

6 

5 

4 

0.000000 

1 

1 

1 

1 

1 

2 

2 

3 

3 

1 

0.6 

0.5 

2 

1.2 

1.0 

3 

1.8 

1.5 

4 

2.4 

2.0 

5 

3.0 

2.5 

6 

3.6 

3.0 

4   n 

1.000000 

9999 

9999 

9999 

9999 

9999 

9998 

9998 

9997 

9997 

7 
8 

4.2 

4.8 

3.5 
4.0 

5.0  n 

9.999996 

96 

95 

95 

95 

95 

95 

95 

95 

95 

9 

5.4 

4.5 

1  n 

95 

94 

94 

94 

94 

94 

94 

94 

93 

93 

2  n 

93 

93 

93 

93 

92 

92 

92 

92 

92 

92 

4 

3 

3  n 

4  n 

91 

89 

91 
89 

91 

89 

91 
88 

90 
88 

90 
88 

90 

87 

90 

87 

90 
87 

89 

87 

1 

0.4 

0.3 

2 

0.8 

0.6  • 

5  n 

9.999986 

86 

86 

85 

85 

85 

84 

84 

83 

83 

3 

1.2 

0.9 

6  n 

83 

82 

82 

81 

81 

81 

80 

80 

79 

79 

4 

1.6 

1.2 

7  n 

78 

78 

77 

77 

76 

78 

75 

74 

74 

73 

5 

2.0 

1.5 

8  n 

73 

72 

71 

71 

70 

69 

69 

68 

67 

til! 

6 

2.4 

1.8 

9  n 

66 

65 

64 

63 

62 

61 

60 

59 

59 

58 

7 

2.8 

2.1 

8 

3.2 

2.4 

6.0  n 

9.999957 

56 

55 

53 

52 

51 

50 

49 

48 

47 

9 

3.6 

2.7 

1  n 

45 

44 

43 

41 

40 

39 

37 

36 

34 

33 

2  n 

31 

30 

28 

26 

25 

23 

21 

19 

17 

15 

2 

1 

3  n 

13 

09 

07 

05 

03 

01 

898 

8% 

893 

4  n 

9.  999891 

888 

886 

883 

880 

878 

875 

872 

869 

866 

1 

0.2 

0.1 

2 

0.4 

0.2 

5  n 

9.999863 

KM 

856 

853 

849 

846 

842 

839 

835 

831 

3 

0.6 

0.3 

6  n 

827 

823 

819 

815 

810 

806 

802 

797 

792 

7S7 

4 

0.8 

0.4 

7  n 

782 

777 

772 

767 

71)1 

756 

750 

744 

738 

732 

5 

1.0 

0.5 

8  n 

726 

720 

713 

706 

700 

693 

685 

678 

671 

663 

6 

1.2 

0.6 

9  n 

655 

647 

639 

631 

622 

613 

604 

595 

585 

576 

7 

1.4 

0.7 

8 

1.1 

0.8 

7.00n 

9.999566 

565 

564 

563 

562 

561 

560 

559 

558 

557 

9 

1.8 

0.9 

170 


U.   S.   COAST   AND   GEODETIC    SURVEY   SPECIAL   PUBLICATION    NO.   H. 

1 


Log  a 

0 

1 

2 

3 

4 

5 

6 

7    8 

9 

Proportional  parts 

7.00  n 

9.  999566 

565 

'564 

563 

562 

561 

560 

559 

558 

557 

01  n 

556 

555 

554 

553 

552 

551 

550 

549 

548 

547 

02  n 

545 

544 

543 

542 

541 

540 

539 

538 

537 

536 

03  n 

535 

534 

533 

532 

531 

530 

528 

527   526 

525 

04n 

524 

523 

522 

521 

520 

519 

517 

516 

515 

514 

05n 

9.  999513 

512 

511 

510 

508 

507 

506 

505 

504 

503 

06n 

502 

501 

499 

498 

497 

4% 

495 

494  ;  492 

491 

07  n 

490 

489 

488 

487 

485 

484 

483 

482 

481 

479 

08  n 

478 

477 

476 

475 

473 

472 

471 

470 

469 

467 

09n 

466 

465 

464 

462 

461 

460 

459 

457 

456 

455 

7.  10  n 

9.  999454 

452 

451 

450 

449 

447 

446 

445 

443 

442 

11  n 

441 

440 

438 

437 

436 

434 

433 

432 

430 

429 

1     2 

12  n 

428 

427 

425 

424 

423 

421 

420 

419 

417 

416 

13  n 

415 

413 

412 

410 

409 

408 

406 

405 

404 

402 

1   0.  1     0.  2 

14  n 

401 

400 

398 

397 

395 

394 

393 

391 

390 

388 

2   0.  2     0.  4 

15  n 

9.  999387 

386 

384 

383 

381 

380 

378 

377 

3/6 

374 

3   0.3     0.6 
4   0.4     0.8 

16  n 

373 

371 

370 

368 

367 

365 

364 

363 

361 

360 

5   0.5      1.0 

17  n 

358 

357 

355 

354 

352 

351 

349 

348 

346 

345 

6   0.  G     1.2 

18  n 

343 

342 

340 

339 

337 

336 

334 

333 

331 

329 

7   0.7     L4 

19  n 

328 

326 

325 

323 

322 

320 

319 

317 

315 

314 

7.20n 

9.999312 

311 

309 

307 

306 

304 

303 

301 

299 

298 

9   0.9     L8 

21  n 

296 

295 

293 

291 

290 

288 

286 

285 

283 

282 

22n 

280 

278 

277 

275 

273 

272 

270 

268 

266 

265 

23n 

263 

261 

260 

258 

256 

255 

253 

251 

249 

247 

24  n 

246 

244 

242 

241 

239 

237 

235 

234 

232 

230 

25n 

9.999228 

227 

225 

223 

221 

219 

218 

216 

214 

212 

26  n 

210 

209 

207 

205 

203 

201 

199 

198 

196 

194 

2',  n 

192 

190 

188 

186 

185 

183 

181 

179 

177 

17:, 

28  n 

173 

171 

169 

168 

166 

164 

162 

160 

158 

1511 

29n 

154 

152 

150 

148 

146 

144 

142 

140 

138 

136 

7.30n 

9.999134 

132 

130 

128 

126 

124 

122 

120 

118 

116 

31  n 

114 

112 

110 

108 

Hid 

104 

102 

100 

09S 

096 

3     4 

32  n 

094 

091 

089 

087 

(is;, 

083 

081 

079 

077   075 

33  n 

072 

070 

C68 

066 

064 

062 

060 

057 

055  !  053 

34  A 

051 

049 

047 

044 

C42 

040 

038 

036 

033  i  031 

1   0.  o     0.  4 
2   0.  6     0.  8 

35  n 

9.999029 

027 

024 

022 

020 

018 

015 

013 

Oil   009 

3   0.9     1.2 

36  n 

006 

004 

002 

8999 

8997 

8995 

8992 

8990 

8988  8985 

4   1.2     1.6 

37  n 

9.998983 

8981 

8978 

8976 

8974 

8971 

8969 

8967 

8964  8962 

5   1.5     2.0 

38  n 

8959 

8957  8955 

8952 

8950 

8947 

8945 

8943 

8940  8938 

6   1.8     2.4 

39  n 

8935 

8933  '  8930 

8928 

8925 

8923 

8920 

8918 

8915  8913 

7   2.  1     2.8 
8   2.4     3.2 

7.40n 

9.998910 

8908 

8905 

8903 

8900 

8898 

8895 

8893 

8890  8888 

9   2.7     3.6 

41  n 

8885 

8883 

8880 

8877 

8875 

8872 

8870 

8867 

8864  8862 

42  n 

8859 

8857 

8854 

8851 

8849 

8846 

8843 

8841 

8838  8835 

43  n 

8833 

8830 

8827 

8825 

8822 

8819 

8816 

8814 

8811  8808 

44  n 

8805 

8803 

8800 

8V97 

8794 

8792 

8789 

8786 

8783 

8781 

45  n 

9.  998778 

8775 

8772 

8769 

8766 

8764 

8761 

8758 

8755 

8752 

46  n 

8749 

8746 

8744 

8741 

8738 

8735 

8732 

8729 

8726 

8723 

47  n 

8720 

8717 

8714 

8711 

8708 

8705 

8702 

8699 

8696 

8693 

48  n 

8690 

8687 

8684 

8681 

8678 

8675 

8672 

8669 

8666 

8663 

49  n 

8660 

8657 

8654 

8651 

8648 

8644 

8641 

8638 

8635 

8632 

7.50n 

9.  998629 

8626 

8622 

8619 

8616 

8613 

8610 

8607 

8603 

8600 

DETERMINATION   OF  AZIMUTH. 
1 


171 


LOKO 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Proportional  parts 

7.  50n 

9.998629 

8626 

8622 

8619 

8616 

8613 

8610 

8607 

8603 

8600 

51  n 

8597 

8594 

8590 

8587 

8584 

8581 

8577 

8574 

8571 

8568 

4 

5 

52  n 

8564 

8561 

8558 

8554 

8551 

8548 

8544 

8541 

8538 

8534 

53  n 

8531 

8528 

8524 

8521 

8517 

8514 

8511 

8507 

8504 

8500 

54  n 

8497 

8493 

8490 

8486 

8483 

8479 

8476 

8472 

8469 

8465 

1 

0.4 

0.5 

2 

0.8 

1.0 

55n 

9.998462 

8458 

8455 

8451 

8448 

8444 

8440 

8437 

8433 

8430 

3 

1.2 

1.5 

56  n 

8426 

8422 

8419 

8415 

8411 

8408 

8404 

8400 

8397 

8393 

4 

1.6 

2.0 

57  n 

8389 

8386 

8382 

8378 

8375 

8371 

8367 

8363 

8360 

8356 

5 

2.0 

2  5 

58  n 

8352 

8348 

8344 

8341 

8337 

8333 

8329 

8325 

8321 

8318 

6 

2.4 

3.0 

59  n 

8314 

8310 

8306 

8302  '  8298 

8294 

8290 

8286 

8282 

8278 

7 

2.8 

3.5 

g 

3.2 

4.0 

7.60  n 

9.998274 

8271 

8267  8263  8259 

8255  8251 

8246 

8242 

8238 

9 

3.6 

4.5 

61  n 

8234 

8230 

8226  ,  8222  8218 

8214  8210 

8206 

82.02 

8197 

62  n 

8193 

8189 

8185 

8181  ,  8177 

8172  8168 

8164 

8160 

8156 

63  n 

8151 

8147 

8143 

8139 

8134 

8130 

8126 

8121 

8117 

8113 

6 

7 

64n 

8108 

8104 

8100 

8095 

8091 

8087 

8082 

8078 

8073 

8069 

65n 

9.998064 

8060 

8055 

8051 

8047 

8042  8038 

8033 

8028 

8024 

0  a 

0  7 

66  n 

8019 

8015 

8010  8006  8001 

7997  ,  7992 

7987 

7983 

7978 

.  O 

67  n 

7973 

7969 

7964  '  7959  :  7955 

7950 

7945 

7941 

7936 

7931 

2 

1.2 
1  ft 

1.  4 
21 

68  n 

7926 

7922 

7917  j  7912  7907 

7902 

7898 

7893 

7888 

7883 

I.  o 

.  1 

69  n 

7878 

7873 

7868 

7863  7859 

7854 

7849 

7844 

7839 

7834 

4 
5 

2.  4 
3.0 

2.8 
3.5 

7.  "On 

9.997829 

7824 

7819 

7814  7809 

7804 

7799 

7794 

7789 

7783 

6 

3.6 

49 

4.2 
J  Q 

71  n 

7778 

7773 

7768 

7763  7758 

7753 

7748 

7742 

7737 

7732 

.  £. 

t.  a 

72  n 

7727 

7722 

7716 

7711  7706 

7700 

7695 

7690 

7685 

7679 

8 

4.8 

5   A 

5.6 
60 

73  n 

7674 

7669 

7663 

7658  7652 

7647 

7642 

7636 

7631 

7625 

.  4 

.  o 

74  n 

7620 

7614 

7609 

7603  7598 

7592 

7587 

7581 

7576 

7570 

75  n 

9.  997565 

7559 

7553 

7548 

7542 

7537 

7531 

7525 

7519 

7514 

<j 

76  n 

7508 

7502 

7497 

7491 

7485 

7479 

7473 

7468 

7462 

7456 

77  n 

7450 

7444 

7438 

7433 

7427 

7421 

7415 

7409 

7403 

7397 

78  n 

7391 

7385 

7379 

7373 

7367 

7361 

7355 

7349 

7343 

7337 

79  n 

7330 

7324 

7318 

7312 

7306 

7300 

7293 

7287 

7281 

7275 

i 

0.8 

0.9 

2 

1.6 

1.8 

7.80  n 

9.  997268 

7262 

7256 

7250 

7243 

7237 

7231 

7224 

7218  7211 

3 

2.4 

2.7 

81  n 

7205 

7199 

7192 

7186 

7179 

7173 

7166 

7160 

7153  7147 

4 

3.2 

3.6 

82  n 

7140 

7134 

7127 

7120 

7114 

7107 

7100 

7094 

7087  7080 

5 

4.0 

4.5 

83n 

7074 

7067 

7060 

7053 

7047 

7040 

7033 

7026 

7019  7013 

6 

4.8 

5.4 

84n 

7006 

6999 

6992 

6985 

6978 

6971 

6964 

6957 

6950  6943 

7 

5.6 

6.3 

8 

6.4 

7.2 

85n 

9.  996936 

6929 

6922 

6915 

6908 

6901 

6894 

6887 

6880  6872 

9 

7.2 

8.1 

86  n 

6865 

6858 

6851 

6844 

6836 

6829 

6822 

68U 

6807  ;  6800 

87  n 

6792 

6785 

6778 

6770 

6763 

6755 

6748 

6740 

6733  6725 

88  n 

6718 

6710 

6703 

6695 

6688 

6680 

6672 

6665 

6657 

6650 

89  n 

6642 

6634 

6626 

6619 

6611 

6603 

6595 

6587 

6580 

6572 

10 

11 

7.90n 

9.996564 

6556 

6548 

6540 

6532 

6524 

6516 

6508 

6500 

6492 

91  n 

6484 

6476 

6468  i  6460 

6452 

6444 

6435 

6427 

6419 

6411 

1 

1.0 

1.1 

92n 

6403 

6394 

6386  6378 

6369 

6361 

6353 

6344 

6336 

6328 

2 

2.0 

2.2 

93  n 

6319 

6311 

6302  ;  6294 

6285 

6277 

6268 

6260 

6251 

6242 

3 

3.0 

3.3 

94  n 

6234 

6225 

6217 

6208 

6199 

6190 

6182 

6173 

6164 

6155 

4 

4.0 

4.4 

5 

5.0 

5.5 

95n 

9.  996146 

6138 

6129 

6120 

6111 

6102 

6093 

6084 

6075 

6066 

6 

6.0 

6.6 

96  n 

6057 

6048 

6039 

6030 

6021 

6012 

6003 

5993 

5984 

5975 

7 

7.0 

7.7 

97  n 

5966 

5956 

5947 

5938 

5929 

5919 

5910 

5900 

5891 

5882 

8 

8.0 

8.8 

98  n 

5872 

5863 

5853 

5844 

5834 

5825 

5815 

5805 

5796 

5786 

9 

9.0 

9.9 

99  n 

5777 

5767 

5757 

5747 

5738 

5728 

5718 

5708 

5698 

56S9 

S.OOn 

9.995079 

5669 

5659 

5649 

5639 

5629 

5619 

5609 

5599 

5589 

172 


U.  S.   COAST  AND  GEODETIC   SURVEY  SPECIAL  PUBLICATION   NO.  14. 

/ 


Logo 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Proportional  parts 

S.OOn 

1995679 

5669 

5659 

5649 

5639 

5629 

5619 

5609 

5599 

5589 

10 

11 

12 

13 

14 

01  n 

5578 

5568 

5558 

5548 

5538 

5528 

5517 

5507 

5497 

5486 

02n 

5476 

5466 

5455 

5445 

5434 

5424 

5413 

5403 

5392 

5382 

1 

1.0 

1.1 

1.2 

1.3 

1.4 

03n 

5371 

5361 

5350 

5339 

5329 

5318 

5307 

5296 

5286 

5275 

2 

2.0 

2.2 

2.4 

2.6 

2.8 

in  n 

5264 

5253 

5242 

5231 

5220 

5209 

5198 

5187 

5176 

5165 

3 

3.0 

3.3 

3.6 

3.9 

4.2 

4 

4.0 

4.4 

4.8 

5.2 

5.6 

05n 

9.995154 

5143 

5132 

5121 

5110 

5098 

5087 

5076 

5065 

5053 

5 

5.0 

5.5 

6.0 

6.5 

7.0 

08n 

5042 

5031 

5019 

5008 

4996 

4985 

4973 

4962 

4950 

4939 

t; 

6.0 

6.6 

7.2   7.8 

8.4 

07  n 

4927 

4916 

4904 

4892 

4<81 

4869 

4857 

4845 

4833 

4822 

7 

7.0 

7.7 

8.4   91 

9.8 

08  n 

4810 

4798 

4786 

4774 

4762 

4750 

4738 

4726 

4714 

4702 

8 

8.0 

8.8 

9.6 

10.4 

11.2 

09n 

4690 

4677 

4665 

4653 

4641 

4628 

4616 

4604 

4591 

4579 

9 

9.0 

9.9 

10.8 

11.7   12.6 

8.  10  n 

9.994567 

4554 

4542 

4529 

4517 

4504 

4492 

4479 

4466 

4454 

15 

16 

17 

18 

19 

11  n 

4441 

4428 

4415 

4403 

4390 

4377 

4364 

4351 

4338 

4325 

12  n 

4312 

4299 

4286 

4273 

4260 

4247 

4234 

4220 

4207 

4194 

t 

1  5 

1  6 

1  7 

18    19 

13  n 

4181 

4167 

41.14 

4141 

4127 

4114 

4100 

4087 

4073 

4080 

2 

3  0 

3  2 

3  4 

3*6  !   38 

14  n 

4046 

4032 

4019 

4005 

3991 

3978 

3964 

3950 

3936 

3922 

3 

4^5 

4^8 

S.I 

5.4  i   5^7 

15  n 

9.  993908 

3894 

3880 

3866 

3852 

3838 

3824 

3810 

3796 

3782 

4 
g 

6.0 
7.5 

6.4 
8.0 

6.8 
8  5 

7.2 
9  0 

7.6 
9.5 

16  n 
17  n 

3767 
3623 

3753 
3609 

3739 
3594 

3725 
3579 

3710 
3565 

3698 
3550 

3681 
3535 

3667 
3521 

3652 

3.106 

3638 
3491 

6 

9^0 
10  5 

9^6 

11  2 

10.3 

11  9 

10.8 

12  6 

11.4 

13  3 

18  n 
19  n 

3476 
3325 

3461 
3310 

3446 
3295 

3431 
3279 

3416 
3264 

3401 
3248 

3386 
3233 

3371 
3218 

3356 
3202 

3340 
3186 

8 
9 

12^0 
13.5 

12!  8 
14.4 

13^6 
15.3 

U.4 

16.2 

15^2 
17.1 

8.20n 

9.993171 

3155 

3140 

3124 

3108 

3092 

3077 

3061 

3045 

3029 

20 

21 

22 

23 

24 

21  n 

3013 

2997 

2981 

2965 

2949 

2933 

2917 

2900 

2884 

2868 

22  n 

2852 

2835 

2819 

2803 

?7SI> 

2770 

2753 

2736 

2720 

2703 

23n 

2687 

2670 

2653 

2636 

2619 

2603 

2586 

2569 

2552 

•J.1.35 

1 

2.0 

2.  1 

2.2 

2.3 

2.  4 

24  n 

2518 

2501 

2483  2466 

2449 

2432  2414 

2397 

2380 

2362 

2 

3 

6.0 

6.3 

6.6 

6.9 

7.2 

25n 

9.992345 

2327 

2310 

2292 

2275 

2257 

2239 

2222 

2204 

2186 

4 

8.0 

8.4 

8.8 

9.2 

9.6 

26  n 

2168 

2150 

2132 

2114 

2096 

2078 

'2060 

2042 

2024 

2008 

5 

10.0 

10.5 

11.0 

11.5 

12.0 

27  n 

1987 

1969 

1951 

1932 

1914 

1896 

1877 

1858 

1840 

1821 

6 

12.0 

12.6 

13.2 

13.8 

14.4 

28  n 

1803 

1784 

1765 

1746 

1727 

1709 

1690 

1671 

1652 

1633 

7 

14.0 

14.7 

15.4 

16.1 

16.8 

29n 

1613 

1594 

1575 

1556 

1537 

1517 

1498 

1478 

1459 

1440 

8 
9 

16.0 
18.0 

16.8 
18.9 

17.6 
19.8 

18.4 
20.7 

19.2 
21.6 

8.30n 

9  991420 

1400 

1381 

1361 

1341 

1322 

1302 

1282 

1262 

1242 

31  n 

1222 

1202 

1182 

1162 

1142 

1122 

1101 

1081 

1061 

1040 

25 

ZD 

27 

28 

29 

32  n 
33  n 

1020 
0813 

0999 
0792 

0979 
0771 

0958 
0750 

0938 
0729 

0917 
0708 

0896 
0886 

0875 
0665 

OS55 
0644 

0834 
0622 

1 

2.5 

2.6 

2.7 

2.8 

2.9 

34  n 

0601 

05SO 

0558 

0537 

0515 

0493 

0472 

0450 

0428 

0406 

2 

5.0 

5.2 

5.4 

5.6 

5.8 

3 

7.5 

7.8 

8.1 

8.4 

8.7 

35n 

9.990385 

0363 

0341 

0319 

0297 

0274 

0252 

0230 

0208 

0186 

4 

10.0 

10.4 

10.8 

11.2 

11.6 

36  n 

0163 

0141 

0118 

0096 

0073 

0051 

0028 

0005 

9982 

8960 

5 

12.5 

13.0 

13.5 

14.0 

14.5 

37  n 

9.  989937 

9914 

9891 

9868 

9845 

9821 

9798 

9775 

9752 

9728 

6 

1.1.  1) 

15.6 

16.2 

16.8 

17.4 

38  n 

9705 

9682 

9658 

9634 

9611 

9587 

9563 

9540 

9516 

9492 

7 

17.5 

18.2 

18.9 

19.6 

20.3 

39  n 

9468 

9444 

9420 

9396 

9372 

9348 

9323 

9299 

9275 

9250 

8 

20.0 

20.8 

21.6 

22.4 

23.2 

9 

22.5 

23.4 

24.3 

25.2 

26.1 

8.40n 

9.  989226 

9201 

9177 

9152 

9127 

9103 

9078 

9053 

9028 

9003 

41  n 

.19  n 

8978 
8725 

8953 

8928 
8673 

8903 
8647 

8877 
8622 

8852 
8596 

8827 
8570 

8801 
8544 

8776 
8518 

8750 
8492 

30    31 

32 

i  -  n 
43n 

8465 

8439 

8413 

8386 

8360 

8334 

8307 

8280 

8254 

8227 

1 

3.0 

3.1 

3.2 

44  n 

8200 

8173 

8147 

8120 

8093 

8066 

8038 

8011 

7984 

7957 

2 

6.0 

6.2 

6.4 

3 

9.0   9.3 

9.6 

45n 

9.  987929 

7902 

7874 

7847 

7819 

7791 

7764 

7736 

7708 

7680 

4 

12.0 

12.4 

12.8 

46n 

7652 

7624 

75% 

7568 

7539 

7511 

7483 

7454 

7426 

7397 

5 

15.0 

15.5 

16.0 

47  n 

7369 

7340 

7311 

7282 

7253 

7224 

7195 

7166 

T137 

7108 

6 

18.0 

18.6 

19.2 

48  n 

7079 

7049 

7020 

6990 

6%1 

6931 

6902 

6872 

6842 

6812 

7 

21.0 

21.7 

22.4 

49  n 

6782 

6752 

6722 

6692 

6662 

6631 

6601 

6571 

6540 

6510 

8 

24.0 

24.8 

25.  6 

9 

27.0 

27.9 

28.8 

8.  50n 

9.  986479 

6448 

6418 

6387 

6356 

6325 

6294 

6263 

6232 

6200 

DETERMINATION    OF   AZIMUTH. 
1 


173 


Log 


I—a 


Logo 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Proportional  parts 

8.50n 

9.986479 

6448 

6418 

6387 

6356 

6325 

ftfll  1 

6294 

IQttrt 

6263 

V  i  is 

6232 
5916 

6200 

CBOJ 

32 

34 

36 

38 

40 

51  n 
52  n 

6169 
5852 

6138 
5820 

6106 
5788 

6075 
5756 

6043 
5723 

OU11 

5691 

OtfoU 

5659 

oy-io 
5626 

5593 

oo&t 
5561 

1 

3.2 

3.4 

3.6 

3.8 

4.0 

53n 

5528 

5495 

5462 

5429 

5396 

5363 

5330 

5297 

5263 

5230 

2 

6.4 

6.8 

7.2 

7.6 

8.0 

54n 

5197 

5163 

5129 

5096 

5(H>2 

5028 

4994 

4960 

4926 

4892 

3 

9.6 

10.2 

10.8 

11.4 

12.0 

4 

12.8 

13.6 

14.4 

15.2 

16.0 

55n 

9.984858 

4823 

4789 

4755 

4720 

4685 

4651 

4616 

4581 

4546 

5 

16.0 

17.0 

18.0 

19.0 

20.0 

56  n 

4511 

4476 

4441 

4406 

4370 

4335 

4300 

4264 

4228 

4193 

6 

19.2 

20.4 

21.6 

22.8 

24.0 

57  n 

4157 

4121 

4085 

4049 

4013 

3977 

3941 

3904 

3868 

3831 

7 

22.4 

23.8 

25.2 

26.6 

28.0 

58  n 

3795 

3758 

3721 

3684 

3648 

3611 

3573 

3536 

3499 

3462 

8 

25.6 

27.2 

28.8 

30.4 

32.0 

59  n 

3424 

3387 

3349 

3312 

3274 

3236 

3198 

3160 

3122 

3084 

8 

28.8 

30.6 

32.4 

34.2 

36.0 

8.60n 
61  n 

9.  983046 
2658 

3007 
2619 

2969 
2580 

2930 
2541 

2892 
2501 

2853 
2462 

2814 
2422 

2776 
2382 

2737 
2343 

2698 
2303 

42 

44 

46 

48 

50 

62  n 
63n 

2263 
1858 

2223 
1817 

2183 
1776 

2142 
1735 

2102 
1694 

2062 
1653 

2021 
1611 

1981 
1570 

1(140 
1528 

1899 
1486 

1 
2 

8.4 

8.8 

9.2 

9.6 

10.0 

64n 

1444 

1403 

1361 

1319 

1276 

1234 

1192 

1149 

1107 

1064 

3 

12.6 

13.2 

13.8 

14.4 

15.0 

4 

16.8 

17.6 

18.4 

19.2 

20.0 

65n 

9.981022 

0979 

0936 

0893 

0850 

0807 

0763 

0720 

0677 

0633 

S 

21.0 

22.0 

23.0 

24.0 

25.0 

66n 

0589 

0546 

0502 

0458 

0414 

0370 

0325 

0281 

0237 

0192 

6 

25.2 

26.4 

27.6 

28.8 

30.0 

67  n 

0147 

0103 

0058 

0013 

5968 

9923 

9878 

9832 

§787 

9741 

7 

29.4 

30.8 

32.2 

33.6 

35.0 

68n 

9.979695 

9650 

9604 

9558 

9512 

9466 

9420 

9373 

9327 

9280 

8 

33.6 

35.2 

36.8 

38.4 

40.0 

69  n 

9234 

9187 

9140 

9093 

9046 

8999 

8952 

8904 

8857 

8809 

9 

37.8 

39.6 

41.4 

43.2 

45.0 

8.70n 

9.978762 

8714 

8666 

8618 

O1OO 

8570 

'-N-1 

8522 

8473 

7QQK 

8425 

7Q'je 

8376 

7CGJ% 

8328 

705ft 

52 

54 

56 

58 

60 

71  n 
72  n 

8279 
7786 

8230 
7736 

8181 
7686 

O1O6 

7636 

INISi 

7586 

8034 
7535 

1000 

7485 

MOO 

7434 

(Sou 

7384 

/SoD 

7333 

1 

5.2 

5.4 

5.6 

5.8 

6.0 

73  n 

7282 

7231 

7180 

7128 

7077 

7026 

6974 

6922 

6870 

6818 

2 

10.4 

10.8 

11.2 

11.6 

12.0 

74  n 

6766 

6714 

6662 

6610 

6557 

6505 

6452 

6399 

6346 

6293 

3 

15.6 

16.2 

16.8 

17.4 

18.0 

4 

20.8 

21.6 

22.4 

23.2 

24.0 

75  n 

9.  976240 

6187 

6133 

6080 

6026 

5972 

5918 

5864 

5810 

5756 

5 

26.0 

27.0 

28.0 

29.0 

30.0 

76  n 

5702 

5647 

5593 

5538 

5483 

5428 

5373 

5318 

5262 

5207 

6 

31.2 

32.4 

33.6 

34.8 

36.0 

77  n 

5152 

5096 

5040 

4984 

4928 

4872 

4816 

4759 

4703 

4646 

7 

36.4 

37.8 

39.2 

40.6 

42.0 

78  n 

4589 

4532 

4475 

4418 

4361 

4304 

4246 

4188 

4131 

4073 

8 

41.6 

43.2 

44.8 

46.4 

48.0 

79  n 

4015 

3957 

3898 

3840 

3781 

3723 

3664 

3605 

3546 

3487 

9 

46.8 

48.6 

50.4 

52.2 

54.0 

8.  80n 

9.973428 

3368 

3309 

3249 

3189 

3129 

3069 

3009 

2949 

2888 

62 

64 

66 

68 

70 

ft!  -n 

9tt98 

97R7 

97flfi 

OftJK 

OKQA 

neno 

9J.fi! 

9-infl 

2338 

o  >7f, 

ol  H 

82n 

dOSIO 

2215 

X/Bf 

2153 

jffUO 

2090 

JO*> 

2028 

*O54 

1966 

IHM0 

1903 

invl 

1840 

^WU 

1777 

1714 

1651 

1 

6.2 

6.4 

6.6 

6.8 

7.0 

83n 

1588 

1525 

1461 

1398 

1334 

1270 

1206 

1141 

1077 

1013 

2 

12.4 

12.8 

13.2 

13.6 

14.0 

84n 

0948 

0883 

0818 

0753 

0688 

0623 

0557 

0492 

0426 

0360 

3 

18.6 

19.2 

19.8 

20.4 

21.0 

4 

24.8 

25.6 

26.4 

27.2 

28.0 

85n 

9.970294 

0228 

0161 

0095 

0028 

9962 

9895 

5828 

9760 

9693 

5 

31.0 

32.0 

33  0 

34.0 

35.0 

86n 

9.969626 

9558 

9490 

9422 

9354 

9286 

9218 

9149 

9081 

9012 

6 

37.2 

38.4 

39.6 

40.8 

42.0 

87  n 

8943 

8874 

8804 

8735 

8666 

85% 

8526 

8456 

8386 

8316 

7 

43.4 

44.8 

46.2 

47.6 

49.0 

88n 

8245 

8175 

8104 

8033 

7962 

7891 

7819 

7748 

7676 

7604 

8 

49.6 

51.2 

52.8 

54.4 

56.0 

89  n 

7532 

7460 

7388 

7316 

7243 

7170 

7097 

7024 

6951 

6878 

9 

55.8 

57.6 

59.4 

61.2 

63.0 

8.90n 

9.966804 

6731 

6657 

coin 

6583 
KCQji 

6509 

C7CQ 

6435 

CftQO 

6360 

^ftft7 

6285 

ecoi 

6211 

c  <  e  t 

6136 

EOyO 

72 

74 

76 

78 

80 

91  n 
92n 

6061 
5301 

5985 
5224 

oyiu 
5147 

OSo4 

5070 

u/oy 
4992 

OOOO 

4915 

OOU/ 

4837 

OOol 

4759 

O-1O4 

4681 

oo/o 
4603 

1 

7.2 

7.4 

7.6 

7.8 

8.0 

93  n 

4525 

4446 

4368 

4289 

4210 

4130 

4051 

3972 

3892 

3812 

2 

14.4 

14.8 

15.2 

15.6 

16.0 

94n 

3732 

3652 

3571 

3491 

3410 

3329 

3248 

3167 

3086 

3004 

3 

21.6 

22.2 

22.8 

23.4 

24.0 

4 

28.8 

29.6 

30.4 

31.2 

32.0 

95n 

9.962922 

2840 

2758 

2676 

2594 

2511 

2428 

2845 

2262 

2179 

5 

36.0 

37.0 

38.0 

39.0 

40.0 

96n 

2095 

2012 

1928 

1844 

1760 

1675 

1591 

1506 

1421 

1336 

6 

43.2 

44.4 

45.6 

46.8 

48.0 

97  n 

1251 

1165 

1080 

0994 

0908 

0822 

0735 

0649 

0562 

0475 

7 

50.4 

51.8 

53.2 

54.6 

56.0 

98n 

9.960388 

0301 

0213 

0126 

0038 

S950 

§862 

5773 

5685 

9596 

8 

57.6 

59.2 

60.8 

62.4 

64.0 

99  n 

9.959507 

9418 

9329 

9239 

9149 

9059 

8969 

8879 

8789 

8698 

9 

64.8 

66.6 

68.4 

70.2 

72.0 

9.00n 

8607 

8516 

8425 

8334 

8242 

8150 

8058 

7966 

7874 

7781 

82 

84 

86 

88 

90 

1 

8.2 

8.4 

8.6 

8.8 

9.0 

2 

16.4 

16.8 

17.2 

17.  i; 

18.0 

3 

24.6 

25.2 

25.8 

26.4 

27.0 

4 

32.8 

33  6 

34.4 

35.2 

36.0 

5 

41.0 

42.0 

43.0 

44.0 

45.0 

6 

49.2 

50.4 

51.6 

52.8 

54.0 

7 

57.4 

58.8 

60.2 

61.6 

63.0 

8 

65.6 

67.2 

68.8 

70.4 

72.0 

9 

73.-S 

75.6 

77.4 

79.2 

81.0 

INDEX. 


Page. 

Additions  to  previous  edition  ....................................  5 

Adjustment  and  description  of  the  transit  micrometer  ............  9 

Adjustment  and  description  of  the  vertical  circle  .................  52 

Adjustments,  direction  method  of  determining  azimuth  ..........  145 

Adjustments  of  the  transit  .......................................  14 

Azimuth  .....................................................  16 

Collimation  ..................................................  15 

Finder  circle  .................................................  16 

Focusing  of  eyepiece  .........................................  14 

Focusing  of  objective  .........................................  14 

Horizontal  axis  ...............................................  15 

Vert  icaiity  of  micrometer  wire  ...............................  15 

Wind  ........................................................  15 

Wire  illumination  ............................................  15 

Adjustments  of  the  zenith  telescope  .............................  106 

Apparatus  for  determining  longitude  by  telegraphic  method, 

arrangement  of  .................................................  81 

Apparent  star  places  for  latitude  work,  computation  of  ...........  116 

Artificial  horizon  .................................................  141 

Azimuth: 

Adjustment  of  transit  for  .....................................  16 

Correction  for  elevation  of  mark  in  computation  of  ............  164 

Correction  for  variation  of  the  pole  in  computation  of  .........  164 

Correction  in  time  computation  ...............................  25 

Curvature  correction  in  computation  of  .......................  150 

Direction  method,  adjustments  ...............................  145 

Direction  method,  computation  of  ............................  148 

Direction  method,  explanation  of  record  and  computation  ____  149 

Discussion  of  errors  .........................................  158 

Example  of  record  and  computation,  direction  method  .......  146 

From  time  observations  ............  ..  .........................  160 

From  time  observations  when  no  transit  micrometer  is  used, 

computation  of  .............................................  163 

From  time  observations  with  the  transit  micrometer,  computa- 

tion of  ......................................................  162 

From  time  observations  with  the  transit  micrometer,  example 

of  record  ...................................................  162 

General  considerations  .......................................  142 

Instruments  ..................................................  139 

Instrument,  shelter  for  .......................................  141 

Instrument  support  ..........................................  139 

Mark  ........................................................  140 

Method  of  repetitions,  computation  of  ........................  154 

Method  of  repetitions,  example  of  record  and  computation.  .  153 

Method  of  repetitions,  explanationof  recordand  computation.  .  155 

Methods  of  determining  astronomic  ...........................  138 

Micrometric  method,  example  of  record  and  computation  .....  155 

Micrometric  method,  explanation  of  record  and  computation.  157 

Observations  made  in  connection  with  triangulation  .........  139 

Primary  .....................................................  138 

Statement  of  costs  ............................................  160 

Summary  of  results  ..........................................  149 

Table  of  log       ..............................................  165 


Books  of  reference  ................................................  5 

Cape  tables,  reduction  mean  to  apparent  declinations  with  .......  Ill 

Care  of  chronometers  .............................................  95 

Chronograph  .....................................................  11 

Chronograph,  electrical  connections  for  ...........................  12 

Chronograpbic  observations  for  tune,  table  of  weights  for  incom- 

plete transits  ...................................................  38 

Chronograph,  use  of  ..............................................  12 

Chronometer  corrections  and  rates  in  longitude  determinations 

with  the  transit  micrometer  ...................................  83 

Chronometers,  care  of  ............................................  95 


Page. 

Chronometers,  comparison  by  coincidence  of  beats.... 96 

Chronomctric  method  of  determining  longitude 95 

Combination  of  results 93 

Computation  of 97 

Discussion  of  errors 100 

Closing  error  in  longitude  between  Key  West  and  Atlanta,  com- 
putation of. 85 

Collimation  adjustment  of  transit 15 

Collimation  axis  of  transit ; 13 

Collimation  correction  in  time  computation 25 

Collimation  of  transit,  line  of 13 

Combination  of  latitude  results,  each  pair  observed  more  than 

once 119 

Combination  of  latitude  results,  when  each  pair  is  observed  but 

once 124 

Comparison  of  chronometers  by  coincidence  of  beats 96 

Complete  least  square  method,  computation  of  time  set  by 41 

Contact  correction  for  transit  micrometer 13 

Correction  for: 

Azimuth  in  time  computation 25 

Collimation  in  time  computation 25 

Curvature  in  azimuth  computation 150 

Curvature  of  apparent  path  of  star  in  computation  of  microme- 
ter value 127 

Differential  refraction  in  latitude  computation 117 

Diurnal  aberration  in  computation  of  time 24 

Elevation  of  mark  in  azimuth  computation 1C4 

Inclination  of  axis  of  transit  in  time  computation 22 

Inequality  of  pivots  of  transit  in  time  computation 23 

Rate  in  time  computation 24 

Variation  of  the  pole  in  azimuth  computation 164 

Variation  of  the  pole  in  latitude  computation 132 

Variation  of  the  pole  in  longitude  computation 85 

Cost  of  azimuth  determinations,  statement  of. 160 

Cost  of  establishing  latitude  station 137 

Cost  of  longitude  determinations,  statement  of. 94 

C  urvature  correction  in  azimuth  computation ISO 

Curvature  of  apparent  path  of  star  in  computation  of  micrometer 

value,  correction  for 127 

Derivation  of  (a.—t)  in  time  computation 25 

Differential  refraction  in  latitude  computation,  correction  for 117 

Differential  refraction  in  latitude  computation,  table  of  correc- 
tions for iig 

Direction  method  for  determining  azimuth 145 

Adjustments 145 

Computation  of 148 

Example  of  record  and  computation 146 

Explanation  of  record  and  computation 149 

Directions  for  observing  latitude 109 

Diurnal  aberration  in  computation  of  time,  correction  for 24 

Diurnal  aberration  in  computation  of  time,  table  of  corrections 

for 24 

Economics  of  latitude  observations 135 

Electrical  connections  for  chronograph 12 

Elevation  of  mark,  correction  to  azimuth  for 164 

Equatorial  intervals  of  transit,  determination  of. 43 

Errors  in  azimuth,  discussion  of 158 

Errors  in  latitude,  discussion  of. 133 

Errors  in  longitude: 

By  chronometric  method,  discussion  of 100 

When  key  and  chronograph  are  used,  discussion  of 93 

When  transit  micrometer  is  used,  discussion  of 85 

Errors  In  time  determinations: 

Discussion  of. 48 

E  sternal 48 

175 


176 


T7.   S.   COAST   AND   GEODETIC   SURVEY   SPECIAL   PUBLICATION    NO.   14. 


Errors  in  time  determinations— Continued.  Page. 

Instrumental 48 

Observer's 50 

Exchange  ot  signals  telegraphic  method  of  determining  Iongitud3, 

record  ol 82 

Eye  and  ear  method  of  observing  time,  directions  lor 19 

Eye  and  ear  observations  Tor  time,  table  of  weights  for  incomplete 

transits 36 

Eyepiece  of  transit,  focusing  of 14 

Finder  circle  adjustment  of  transit 16 

Focusing  of  eyepiece  of  transit 14 

Focusing  of  objective  of  transit 14 

Horizontal  axis  of  transit,  adjustment  of 15 

Illumination  of  wires  of  transit 15 

Inclination  of  axis  of  transit  in  time  computation,  correction  for . .  22 
Incomplete  transits: 

In  chronographic  observations  for  time,  table  of  weights  for —  38 

In  eye  and  ear  observations  for  time,  table  of  weights  for 36 

In  time  computation,  reduction  of 32 

Table  for  use  in  computation  of 32 

With  transit  micrometer 24 

Inequality  ot  pivots  of  transit  in  time  computation,  correction  for.  23 

Inequality  of  pivots  ol  transit,  determination  of 44 

Instructions  for  determining  longitude  with  the  transit  micrometer 

in  high  latitudes 80 

Instructions  for  determining  longitude  with  the  transit  micrometer 

in  Jow  latitudes 79 

Instructions  lor  latitude  work,  general 103 

Key  method  of  observing  time,  computation  of  transit  obser- 
vations    30 

Key  method  of  observing  time,  directions  for 18 

Latitude: 

Combination  of  results,  each  pair  observed  more  than  once ...  119 

Combination  ol  results  when  each  pair  is  observed  but  once . .  124 

Computation 112 

Computation  of  apparent    tarplaces 116 

Computation  oi   value  Ji  micrometer  from  observations  on 

a  close  circumpolar  star 126 

Correction  for  curvature  ol  apparent  path  of  star  in  computa- 
tion of  micrometer  value 127 

Correction  for  differential  refraction 117 

Cost  of  establishing  station 137 

Determination  of  level  and  micrometer  values 124 

Determination  of  micrometer  value  from  observations  of 129 

Directions  for  observing 109 

Discussion  of  errors 132 

Economics  of  observations  for 135 

Example  of  record  and  computation Ill 

Explanation  of  computation 115 

From  a  single  pair,  weight  to  be  assigned  to  mean 135 

General  instructions  for  determining 103 

General  notes  on  computation  of 115 

Methods  of  determining 103 

Observing  list  (form  1) 108 

Observing  list  (form  2) 109 

Reduction  for  variation  of  pole 132 

Reduction  mean  to  apparent  declinations  with  Cape  tables. . .  Ill 

Reduction  to  sea  level 130 

Reduction  to  the  meridian 119 

Summary  of  computation 114 

Table  for  reduction  to  sea  level 131 

Table  of  corrections  for  differential  refraction 118 

Table  of  corrections  for  reduction  to  the  meridian 119 

Level  and  micrometer  values,  determination  of 124 

Level  value  of  transit,  determination  of 46 

Line  intervals  for  transit  No.  18,  table  of 33 

Line  of  collimation  of  transit 13 

Longitude: 

Arrangement  of  apparatus,  telegraphic  method  of  determining  81 

By  wireless  telegraphy 78 

Chronometer  corrections  and  rates,  In  determination  of 83 

Cnronometric  method,  computation  of 97 

Combination  of  results  by  chronometric  method 98 

Combination  of  results  when  no  transit  micrometer  is  used ...  89 


Longitude— Continued.  Page, 

Computation  of  closing  error  between  Key  West  and  Atlanta.  85 

Computation  of  difference,  when  transit  micrometer  is  used ...  84 

Correction  for  variation  of  the  pole 85 

Determination,  computation  when  no  transit  micrometer  is 

used 

Determination,  program  when  no  transit  micrometer  is  used . .  87 

Determination,  statement  of  cost 94 

Discussion  of  errors  in  chronometric  method  of  determining . .  100 

Discussion  of  errors  when  key  and  chronograph  are  used 93 

Discussion  of  errors  when  transit  micrometer  is  used 85 

Instructions  for  use  of  the  transit  micrometer  in  high  latitudes 

for  determining 80 

Instructions  for  the  use  of  the  transit  micrometer  in  low  lati- 
tudes for  determining 79 

Method  of  operations  when  transit  micrometer  is  used 81 

Program  and  apparatus  of  the  telegraph  ic  method 79 

Record  of  exchange  of  signals,  telegraphic  method  of  determin- 
ing   82 

Three  general  methods  of  determining 78 

Weights  assigned  to  separate  chronometers  in  chronometric 

method  of  determining 100 

Mark  for  azimuth  observations 140 

Meridian  telescope,  description  of 8 

Method  of  operations  for  determining  longitude,  transit  micrometer 

method SI 

Methods  of  determining  astronomic  azimuth 138 

Methods  of  determining  latitude 103 

Micrometer  and  level  values,  determination  of 124 

Micrometer,  transit 8 

Micrometer  value  from  latitude  observations,  determination  of —  129 
Micrometer  value  from  observations  on  a  close  circumpolar  star, 

computation  of 126 

Micrometer  wire  of  transit,  test  of  verticality  of 15 

Micrometric  method  of  determining  azimuth,  example  of  record 

and  computation loo 

Micrometric  method  of  determining  azimuth,  explanation  of  rec- 
ord and  computation 157 

Notes  on  computation  of  latitude,  general 115 

Objective  of  transit,  focusing  of 14 

Observatories  and  observing  tents 105 

Observing  for  determination  of  time,  directions  for 18 

Observing  list  for  determination  of  time 17 

Observing  list  (form  1)  for  latitude 108 

Observing  list  (form  ?)  for  latitude 109 

Parallax,  table  of  sun's 60 

Personal  equation  in  time  determination 90 

Personal  equation  in  time  determination,  table  of  relative 92 

Pivot  inequality  of  transit,  determination  of 44 

Pointing  lines 141 

Pole  variation  in  azimuth  computation,  correction  for 164 

Pole  variation  in  latitude  computation,  correction  for 132 

Pole  variation  in  longitude  computation,  correction  for 85 

Primary  azimuth 138 

Rate  correction  in  time  computation 24 

Record  and  computation: 

Direction  method  of  determining  azimuth,  example  of 146 

For  determination  of  time,  example  of 20 

Micrometric  method  of  determining  azimuth,  example  of 155 

Of  latitude  determination,  example  of Ill 

Of  time  by  the  second  method,  example  of 28 

Repetition  method  of  determining  azimuth,  example  of 153 

Record,  azimuth  from  time  observations  with  the  transit  microme- 
ter, example  of 162 

Record  of  observations  on  stars  with  the  vertical  circle  for  determi- 
nation of  time 54 

Record  of  observations  on  the  sun  with  the  vertical  circle  for  deter- 
mination of  time 56 

Reduction  mean  to  apparent  declinations  with  Cape  tables Ill 

Reduction  to  the  meridian  in  latitude  computation 119 

Reduction  to  the  meridian  in  latitude  computation,  table  of  correc- 
tions for 119 

Reference  books 5 

Refraction,  correction  for  differential 117 


INDEX. 


177 


Page. 

Refraction  tables 5S 

Repetition  method  of  determining  azimuth: 

Computation  of 154 

Example  of  record  and  computation 153 

Explanation  of  record  and  computation 155 

Sea  level  reduction  for  latitude 130 

Sextant  observations  lor  time 52 

Shelter  for  azimuth  instrument 141 

Star  factors  for  use  in  computation  of  time 60 

Star  factors  obtained  graphically 61 

Star  factors,  table  ot 62 

Star  list  for  time  determinations 29 

Star  observatio'ns  with  the  vertical  circle  to  determine  time 53 

Stars  for  time  observations,  selection  of 42 

Striding  level  of  transit,  adjustment  of 15 

Sun  observations  with  transit  to  determine  time 51 

Sun  observations  with  vertical  circle  to  determine  time 56 

Sun's  parallax,  table  of 60 

Support  for  latitude  instrument 105 

Supports  for  azimuth  instrument 139 

Tables  (see  list  of  tables  on  p.  4). 

Telegraphic  method  of  determining  longitude,  program  and  appa- 


ratus. 


79 

Tents  and  observatories,  observing 105 

Time: 

By  means  of  the  transit  instrument 7 

Collimation  correction  in  computation  of 25 

Computation  of  observations  on  stars  with  vertical  circle  to 

determine 55 

Computation  of  observations  on  the  sun  with  vertical  circle  to 

determine 56 

Computation  of  transit  observations  for 21 

Computation  of  transit  observations,  key  method  of  observing.  30 

Correction  for  azimuth  in  computation  of 25 

Corrrections  for  diurnal  aberration  in  computation  of 24 

Derivation  of  (ct — t)  in  computation  of 25 

Directions  for  observing  by  eye  and  ear  method 10 

Directions  for  observing  by  key  method 18 

Directions  for  observing  by  transit  micrometer  method 18 

Directions  for  observing  for  determination  of 18 

Discussion  of  errors  in  determination  of 48 

Example  of  record  and  computation  for  determination  of 20 

Example  of  record  and  computation,  second  method 28 

External  errors  in  determination  of 48 

Instrumental  errors  in  determination  of 48 

Observations,  azimuth  from 160 

Observations  on  the  sun  with  transit  to  determine 51 

Observers  errors  in  determination  of 50 

Observing  list  for  determination  of 17 

Other  methods  of  determining 51 

Personal  equation  in  determination  of 90 

Rate  correction  in  computation  of 24 

Record  of  observations  on  stars  with  vertical  circle  to  deter- 
mine   54 

Record  of  observations  on  the  sun  with  vertical  circle  to  de- 
termine   56 

Reduction  of  incomplete  transits  in  computation  of 32 

Relative  weights  depending  on  star's  declination  in  computa- 
tion of 38 

Selection  of  stars  for  observations  of 42 

Set,  computation  by  complete  least  square  method 41 

Set,  computation  by  least  square  method 39 

Set,  explanation  of  second  method  of  computation  of 34 

Set,  explanation  of  usual  method  of  computation  of 27 

Set,  second  method  of  computation  of 34 

Set,  usual  method  of  computation  of 26 

Sextant  observations  for 52 

Star  factors  for  use  in  computation  of 60 

Star  list  for  determination  of 29 

Table  for  use  in  computing  incomplete  transits  in  computa- 
tion of 32 

Table  of  corrections  for  diurnal  aberration  in  computation  of.  24 

8136°— 13 12 


Page. 

T  ime — Continued. 

Table  ot  relative  personal  equation  In  determination  of 92 

Table  of  star  factors  tor  use  in  computation  ol 61 

Table  ot  weights  to  transits  depending  on  the  star's  decima- 
tion in  computation  ol 39 

Vertical  circle  observations  tor 52 

Vertical  circle  observations  on  a  star  to  determine 53 

Vertical  circle  observations  on  the  sun  to  determine 56 

Weights  for  incomplete  transits  in  chronographic  observations 

for 38 

Weights  for  incomplete  transits  in  eyo  and  ear  observations  for.  36 

Transit,  adjustments  of: 

Azimuth 16 

Collimation 15 

Tinder  circle 16 

Focusing  of  eyepiece 14 

Focusing  of  objective 14 

Horizontal  axis 15 

Verticality  of  micrometer  wires 15 

Wind 15 

Wire  illumination 15 

Transit: 

Broken  telescope 8 

Collimation  axis  of 13 

Correction  for  inclination  of  axis  of 22 

Correction  for  inequality  of  pivots  of 23 

Description  of  large  portable 7 

Determination  of  equatorial  intervals  of 43 

Determination  of  level  value  of 46 

Determination  of  pivot  inequality  of 44 

Instrument,  determination  of  time  by  means  of 7 

Line  of  Collimation  of 13 

Micrometer 8 

Micrometer,  contact  correction  for 13 

Micrometer,  description  and  adjustment 9 

Micrometer,  incomplete  transits  with 24 

Micrometer  method  of  observing  time,  directions  for 18 

Observations  for  time,  computation  of 21 

Observations,  key  method  of  observing  time,  computation  of.  30 

Observations  on  the  sun  to  determine  time 51 

Triangulation,  azimuth  observations  made  in  connection  with 139 

Variation  of  pole  in  azimuth  computation,  correction  for 164 

Variation  of  pole  in  latitude  computation,  correction  for 132 

Variation  of  pole  in  longitude  computation,  correction  for 85 

Vertical  circle: 

Computation  of  time  from  observations  on  stars  with 55 

Computation  of  time  from  observations  on  the  sun  with 56 

Description  and  adjustments 52 

Observations  for  time 52 

Record  of  observations  on  stars  for  determination  of  time  with.  54 
Record  of  observations  on  the  sun  for  determination  of  time 

with 56 

Time  from  observations  on  a  star  with 53 

Verticality  of  micrometer  wire  of  transit,  test  of 15 

Weights: 

Assigned  to  separate  chronometers  in  longitude  determination 

by  chronometric  method 100 

Assigned  to  separate  chronometers  in  longitude  determination 

by  chronometric  method,  computation  of 100 

Depending  on  star's  declination  in  time  computation,  relative.  38 
For  incomplete  transits  in  chronographic  observations  for 

time,  table  of 38 

For  incomplete  transits  in  eye  and  ear  observations  for  time, 

table  of 36 

To  be  assigned  to  mean  latitude  from  a  single  pair 135 

To  transits  depending  on  the  star's  declination,  table  of 39 

Wind  adjustment  of  transit 15 

Wireless  telegraphy,  longitude  by 78 

Xonith  telescope,  adjustments  of 106 

Zenith  telescope,  description  of 104 


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